Exponents, Roots and Scientific Notation: Lesson Test

Answer: C

Correct answer: 81. The exponent says to use 3 as a factor four times:

• A: This adds the base and the exponent, 3 + 4.
• B: This is $4^3$; the base and exponent were swapped.
• C (correct): 3 × 3 × 3 × 3 = 81.
• D: This multiplies the base by the exponent, 3 × 4, instead of repeating the factor.

Answer: C

Correct answer: x⁸. The bases match, so keep the base and add the exponents:

• A: This subtracts the exponents, which is the rule for dividing powers.
• B: This multiplies the exponents, which is the rule for a power of a power, not a product.
• C (correct): Same base, so the exponents add: 5 + 3 = 8.
• D: The exponent is right, but the base stays x; nothing doubles the coefficient.

Answer: A

Correct answer: 12. Since

the principal square root is 12. The radical sign means the nonnegative root, so −12 is not the value.

• A (correct): 12 × 12 = 144, and the radical sign gives the nonnegative root.
• B: This divides 144 by 2 instead of finding a number that times itself gives 144.
• C: (−12)² is 144, but the radical sign means the principal, nonnegative root.
• D: 14 squared is 196, not 144.

Answer: D

Correct answer: 4.5 × 10⁻⁴. Move the decimal point right until the front number is between 1 and 10: 4.5. That took 4 moves, and the number is small, so the exponent is −4: $4.5\times 10^{-4}$.

• A: This equals 0.00045, but the front number 45 is not between 1 and 10, so it is not scientific notation.
• B: This equals 0.0045; the decimal point moved one place too few.
• C: A positive exponent makes 45,000; small numbers need a negative exponent.
• D (correct): The front number 4.5 is between 1 and 10, and a small number needs a negative exponent of 4 places.

Answer: B

Correct answer: 1/16. A negative exponent means the reciprocal with a positive exponent:

• A: A negative exponent makes a fraction, not a negative number.
• B (correct): Take the reciprocal: 1 over $4^2$, which is 1/16.
• C: This uses 4 × 2 = 8 in the denominator instead of $4^2=16$.
• D: This multiplies 4 by −2 and treats the result as negative.

Answer: D

Correct answer: 16x¹². The outside exponent applies to every factor:

The result is $16x^{12}$.

• A: This multiplies 2 by 4 instead of raising 2 to the fourth power.
• B: This adds the exponents 3 and 4; a power of a power multiplies them.
• C: The power was applied to x but not to the 2 inside the parentheses.
• D (correct): $2^4=16$ and the exponents multiply, 3 × 4 = 12.

Answer: C

Correct answer: a⁴. Power of a power first:

Then divide by subtracting exponents:

• A: This adds the exponents 10 and 6 instead of subtracting them when dividing.
• B: This adds 2 + 3 for the power of a power, making the denominator $a^5$.
• C (correct): The denominator is $a^6$, and 10 − 6 = 4.
• D: This subtracts the top exponent from the bottom one, 6 − 10, reversing the quotient rule.

Answer: D

Correct answer: 5√3. Split 75 into a perfect square times what remains:

• A: This uses 75 = 5 × 15 and pulls out 5 even though 5 is not a perfect square.
• B: This places the factors backward; $3\sqrt{5}$ equals $\sqrt{45}$, not $\sqrt{75}$.
• C: The 25 was pulled out without taking its square root.
• D (correct): 75 = 25 × 3, and $\sqrt{25}=5$ comes out of the radical.

Answer: B

Correct answer: 7 and 8. Bracket 55 between perfect squares:

Since

the root of 55 is between 7 and 8.

• A: This halves 55 instead of taking its square root.
• B (correct): 49 < 55 < 64, so the root is between 7 and 8.
• C: These bracket roots of numbers from 36 to 49, and 55 is above 49.
• D: These are the square roots of 25 and 36; 55 is much larger than 36.

Answer: C

Correct answer: 1.4 × 10⁴. Multiply the front numbers:

Add the exponents:

That gives $14\times 10^3$, and moving to a front number below 10 gives $1.4\times 10^4$.

• A: This adds the front numbers 4 and 3.5 instead of multiplying them.
• B: The front number was changed from 14 to 1.4 without raising the exponent to keep the value the same.
• C (correct): $14\times 10^3$ is rewritten with front number 1.4 by raising the exponent by one.
• D: This multiplies the exponents 5 and −2 instead of adding them.

Answer: D

Correct answer: −2. Cube roots of negative numbers are real:

because

Also

• A: This makes both cube roots negative, adding −5 and −3.
• B: This swaps the signs of the two roots, using 5 + (−3); the cube root of −125 is the negative one.
• C: This ignores the negative sign and adds 5 + 3.
• D (correct): The cube roots are −5 and 3, and −5 + 3 = −2.

Answer: B

Correct answer: 4 × 10⁴. Divide the fronts:

Subtract the exponents:

That gives $0.4\times 10^5$. Moving to a front number of 4 lowers the exponent by one: $4\times 10^4$.

• A: The front number was changed from 0.4 to 4 without lowering the exponent, making the value ten times too large.
• B (correct): $0.4\times 10^5$ becomes $4\times 10^4$ when the front number is raised to 4.
• C: This raises the exponent when changing 0.4 to 4; making the front larger requires lowering the exponent.
• D: This adds the exponents −4 and −9 instead of subtracting them, then adjusts the front.

Answer: A

Correct answer: −4. Same base, so add exponents:

Any nonzero base to the zero power is 1, and

• A (correct): $2^2=4$, $6^0=1$ and $(-3{)}^2=9$, so 4 + 1 − 9 = −4.
• B: This treats $(-3{)}^2$ as −9, so subtracting it adds 9.
• C: This treats $6^0$ as 0 instead of 1.
• D: This treats $6^0$ as 6 instead of 1.

Answer: A

Correct answer: 1.6 × 10⁶. Divide the capacity by the size of one photo. Divide the fronts:

Subtract the exponents:

The drive holds $1.6\times 10^6$ photos.

• A (correct): 4 ÷ 2.5 = 1.6 and 12 − 6 = 6.
• B: This multiplies capacity by photo size instead of dividing.
• C: This divides 2.5 by 4, flipping the front numbers, which gives 0.625 instead of 1.6.
• D: This divides the exponents, 12 ÷ 6, instead of subtracting them.

Answer: B

Correct answer: (3a²)/b². Apply the outside power to every factor:

Divide:

• A: This adds exponents for $(a^2{)}^3$, getting $a^5$ instead of $a^6$.
• B (correct): The numerator becomes $27a^6{b}^3$; dividing gives 3, $a^2$ and $b^{-2}$.
• C: This leaves the 3 uncubed, so the coefficient becomes 3 ÷ 9 = 1/3.
• D: This subtracts the b exponents backward; $b^{3-5}=b^{-2}$ puts $b^2$ in the denominator.

Answer: A

Correct answer: y⁵. Multiply exponents for the power of a power:

Add exponents when multiplying:

Subtract when dividing:

• A (correct): Power of a power gives y^12, multiplying by y^-2 gives y^10, and dividing by y^5 leaves y^5.
• B: This treats the -2 exponent as +2, giving 12 + 2 - 5 = 9.
• C: This adds the bottom exponent instead of subtracting it: 12 - 2 + 5 = 15.
• D: This adds 3 and 4 for the power of a power, getting y^7, so 7 - 2 - 5 = 0 and y^0 = 1.

Answer: C

Correct answer: 2 × 10⁷. Multiply the top: 6 times 3 is 18 and

so the top is $18\times 10^3$. Divide by the bottom: 18 divided by 9 is 2, and

The result is $2\times 10^7$, already in proper form.

• A: This adds the bottom exponent instead of subtracting it: 3 + (-4) = -1.
• B: This drops the minus sign on 10^-2, so the top exponent becomes 7 and 7 + 4 = 11.
• C (correct): The top is 18 x 10^3; dividing gives 2 and an exponent of 3 - (-4) = 7.
• D: This subtracts in the wrong order, taking -4 - 3 instead of 3 - (-4).

Answer: C

Correct answer: -63. Without parentheses the minus sign is not part of the base, so

With parentheses the base is negative, and an odd power keeps the sign:

• A: This reads -6^2 as (-6)^2 = 36, then adds -27.
• B: This gets -36 correctly but treats (-3)^3 as +27, as if an odd power of a negative were positive.
• C (correct): Only 6 is squared in -6^2, giving -36, and (-3)^3 = -27, so the sum is -63.
• D: This makes both powers positive, 36 + 27, ignoring both negative signs.

Answer: D

Correct answer: 6√2. Pull out perfect-square factors:

All three are multiples of $\sqrt{2}$, so combine the numbers in front:

giving $6\sqrt{2}$.

• A: This adds and subtracts the numbers under the radicals, 50 + 18 - 8, which is not allowed.
• B: This simplifies correctly but adds the last term instead of subtracting it: 5 + 3 + 2.
• C: This combines the first two terms but leaves out the subtraction of root 8, which is 2 root 2.
• D (correct): Each radical simplifies to a multiple of root 2: 5, plus 3, minus 2, gives 6 root 2.

Answer: B

Correct answer: 4y⁶. Treat the numbers and the variable separately. Divide the coefficients:

For the same base, subtract exponents:

The result is $4y^6$.

• A: This subtracts the coefficients, 24 - 6, instead of dividing them.
• B (correct): Divide the coefficients, 24 / 6 = 4, and subtract the exponents, 10 - 4 = 6.
• C: This adds the exponents, which is the rule for multiplying, not dividing.
• D: This multiplies the exponents, which is the rule for a power of a power.

Answer: C

Correct answer: 1/27. Rewrite with a positive exponent in the denominator:

Since

the value is $\frac{1}{27}$, a small positive fraction.

• A: This treats the negative exponent as making the number negative, which it never does.
• B: This multiplies the base by the exponent, 3 times -3, instead of using repeated factors.
• C (correct): A negative exponent means the reciprocal: 1 over 3^3, which is 1/27.
• D: This takes the reciprocal but squares 3 instead of cubing it.

Answer: D

Correct answer: 64. An exponent counts repeated factors:

Doubling step by step gives 2, 4, 8, 16, 32, 64.

• A: This multiplies the base by the exponent, 2 times 6, instead of repeating the factor.
• B: This uses 2 as a factor only five times, which is 2^5.
• C: This uses 2 as a factor seven times, which is 2^7.
• D (correct): Using 2 as a factor six times gives 2, 4, 8, 16, 32, 64.

Answer: B

Correct answer: 0.000725 m. For a negative power of ten, move the decimal point left by the size of the exponent. Starting at 7.25 and moving four places left gives 0.000725 m.

• A: This moves the decimal point only three places to the left.
• B (correct): A negative exponent of 4 moves the decimal point four places to the left.
• C: This moves the decimal point five places to the left, one too many.
• D: This moves the decimal point to the right, which is only done for a positive exponent.

Answer: C

Correct answer: 1. By the zero-exponent rule,

for any nonzero a. One way to see it:

and any nonzero number divided by itself is 1.

• A: This confuses a zero exponent with multiplying by zero.
• B: This treats a zero exponent as leaving the base unchanged.
• C (correct): Any nonzero number raised to the zero power equals 1.
• D: This is 18 to the power -1, not to the power 0.

Answer: B

Correct answer: 7√2. Find the largest perfect-square factor of 98, which is 49:

Check:

• A: This swaps the parts: it takes 2 outside and leaves 7 inside, which equals the root of 28.
• B (correct): 98 = 49 x 2, and the root of 49 is 7, leaving root 2.
• C: This pulls out 49 itself instead of its square root, 7.
• D: This takes out 7 but then leaves 14 inside; 7 root 14 equals the root of 686.

Answer: D

Correct answer: 11.4 ft. The side of a square is the square root of its area. Since

$\sqrt{130}$ is between 11 and 12, and closer to 11. Only 11.4 ft fits, and

• A: This is below 11, but 11 squared is only 121, which is already less than 130.
• B: This is above 12, but 12 squared is 144, which is already more than 130.
• C: This divides the area by 2 instead of taking the square root.
• D (correct): 130 lies between 121 and 144, so the side is between 11 and 12 ft, and 11.4 squared is about 130.

Answer: A

Correct answer: 6. A cube root asks which number, used as a factor three times, gives 216. Try small numbers:

• A (correct): 6 x 6 x 6 = 216, so 6 is the cube root.
• B: This divides 216 by 3 instead of finding a number used as a factor three times.
• C: This is 6 squared, which is the result of using 6 as a factor only twice.
• D: 8 cubed is 512, far more than 216.

Answer: C

Correct answer: 125 cu ft. Cubed means the base is used as a factor three times:

• A: This multiplies the base by the exponent, 5 times 3.
• B: This squares 5 instead of cubing it.
• C (correct): 5 x 5 = 25, and 25 x 5 = 125.
• D: This uses 5 as a factor four times, which is 5^4.

Answer: C

Correct answer: -8a⁹b⁶. An outside exponent applies to every factor:

Together that is $-8a^9{b}^6$.

• A: This drops the sign; an odd power of a negative number stays negative.
• B: This multiplies -2 by 3 instead of cubing it.
• C (correct): Cube each factor: (-2)^3 = -8, (a^3)^3 = a^9 and (b^2)^3 = b^6.
• D: This adds the outside exponent to each inside exponent instead of multiplying.

Answer: A

Correct answer: 500 s. Time = distance divided by speed:

Rewrite 0.5 as $5\times 10^{-1}$: the result is

a little over 8 minutes.

• A (correct): Time is distance divided by speed: 1.5 / 3 = 0.5 and 11 - 8 = 3, giving 0.5 x 10^3 = 500 s.
• B: This divides speed by distance, getting 2 x 10^-3, and then reads the negative exponent as positive.
• C: This shifts the decimal the wrong way when fixing 0.5 x 10^3, writing 5 x 10^1 instead of 5 x 10^2.
• D: This writes the front as 5 but keeps the exponent at 3, making the answer ten times too big.

Answer: A

Correct answer: 6. Handle each power: without parentheses

with parentheses

because the even power makes it positive; and

• A (correct): -3^2 = -9, (-2)^4 = 16 and 4^0 = 1, so -9 + 16 - 1 = 6.
• B: This reads -3^2 as (-3)^2 = 9, then computes 9 + 16 - 1.
• C: This treats 4^0 as 0 instead of 1, giving -9 + 16 - 0.
• D: This makes (-2)^4 negative; an even power of a negative number is positive.

Answer: D

Correct answer: 3.2 × 10⁴ kg. Multiply the front numbers and add the exponents:

giving $32\times 10^3$. Since 32 is 10 or more, rewrite it as $3.2\times 10^4$ kg.

• A: This changes 32 to 3.2 but forgets to raise the exponent by one.
• B: This adds the front numbers, 4 + 8, instead of multiplying them.
• C: This multiplies the exponents, 6 times -3, instead of adding them.
• D (correct): 32 x 10^3 is rewritten as 3.2 x 10^4 by moving the decimal and raising the exponent by one.

Answer: B

Correct answer: 4.7 × 10⁷ acres. Place the decimal after the first nonzero digit to get 4.7, which is between 1 and 10. Count how many places it moved from the end of 47,000,000: seven places, so the number is $4.7\times 10^7$.

• A: This counts only six places; 4.7 x 10^6 is 4,700,000.
• B (correct): The decimal point moves seven places left to give a front number of 4.7.
• C: This has the right value but is not scientific notation, because the front number 47 is not less than 10.
• D: A negative exponent is used for small numbers, not for a large number like this.

Answer: D

Correct answer: 3⁵. Product rule: with the same base, keep the base and add the exponents,

Check:

• A: This multiplies the exponents, which is the rule for a power of a power.
• B: This multiplies the bases as well as adding the exponents; the base stays 3.
• C: This multiplies both the bases and the exponents.
• D (correct): Same base, so keep 3 and add the exponents: 2 + 3 = 5.

Answer: B

Correct answer: 2.016 × 10⁷ km. Convert days to hours:

Distance = speed times time:

Rewrite with a front number below 10: $2.016\times 10^7$ km.

• A: This gets the front number right but drops one power of ten when rewriting 20.16 x 10^6.
• B (correct): Hours in 30 days are 720, and 2.8 x 10^4 x 720 = 20,160,000 = 2.016 x 10^7 km.
• C: This adds one power of ten too many when moving the decimal point.
• D: This multiplies the hourly speed by 30 days, forgetting to convert days to hours.

Answer: D

Correct answer: 2¹⁰. Evaluate each power as repeated multiplication:

The largest is $2^{10}$. A large exponent on a small base can beat a large base with a small exponent.

• A: 10^3 = 1,000, slightly less than 2^10 = 1,024.
• B: 3^6 = 729, smaller than both 1,000 and 1,024.
• C: 5^4 = 625, the smallest of the four values.
• D (correct): 2^10 = 1,024, which beats 1,000, 729 and 625.

Answer: A

Correct answer: ab⁵. Square every factor in the numerator:

Divide by the denominator:

The result is $a{b}^5$.

• A (correct): The top is 4a^-4 b^6; dividing gives 4/4 = 1, a^(-4 - (-5)) = a^1 and b^(6 - 1) = b^5.
• B: This subtracts -5 as if it were +5, taking -4 - 5 = -9 for the exponent of a.
• C: This forgets to square the coefficient 2, so it divides 2 by 4.
• D: This adds 3 and 2 for the power of b instead of multiplying, giving b^5 on top and b^4 after dividing.

Answer: A

Correct answer: 169. A perfect square is a whole number times itself. Squares near these values are

Only 169 is on that list.

• A (correct): 169 = 13 x 13, so it is the square of a whole number.
• B: 160 lies between 144 (12 squared) and 169 (13 squared), so it is not a perfect square.
• C: 150 also lies between 144 and 169, so its square root is not a whole number.
• D: 125 is a perfect cube (5 cubed), but it lies between 121 and 144, so it is not a perfect square.

Answer: B

Correct answer: 5/4. Use the zero rule,

and the negative-exponent rule,

Add them:

which is 1.25.

• A: This treats 4^0 as 0 instead of 1.
• B (correct): 4^0 = 1 and 4^-1 = 1/4, so the sum is 1 and 1/4, or 5/4.
• C: This reads 4^-1 as -4, making the negative exponent a negative number.
• D: This ignores the negative sign in the exponent, using 4^1 = 4.