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PSAT Math : Exponential Operations

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Example Questions

Example Question #3 : How To Add Exponents

If $3^{4x+6}=27^{2x}$ , what is the value of ?

Possible Answers:

Correct answer:

Explanation:

Using exponents, 27 is equal to 3³. So, the equation can be rewritten:

3^{4x + 6} = (3³)^2x

3^{4x + 6} = 3^6x

When both side of an equation have the same base, the exponents must be equal. Thus:

4x + 6 = 6x

6 = 2x

x = 3

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Example Question #81 : Exponential Operations

What is the value of such that 2^n=4^2*16^3*32^2 ?

Possible Answers:

480

240

Correct answer:

Explanation:

We can solve by converting all terms to a base of two. 4, 16, and 32 can all be expressed in terms of 2 to a standard exponent value.

$4=2^2\rightarrow 4^2=(2^2)^2$

$16=2^4\rightarrow 16^3=(2^4)^3$

$32=2^5\rightarrow 32^2=(2^5)^2$

We can rewrite the original equation in these terms.

2^n=(2^2)^2*(2^4)^3*(2^5)^2

Simplify exponents.

$2^n=(2^4)*(2^{12})*(2^{10})$

Finally, combine terms.

$2^n=2^{4+12+10}=2^{26}$

From this equation, we can see that n=26 .

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Example Question #13 : How To Add Exponents

How many of the following base ten numbers have a base five representation of exactly four digits?

(A) 100

(B) 200

(D) 400

Possible Answers:

Four

Three

Two

One

None

Correct answer:

Three

Explanation:

A number in base five has powers of five as its place values; starting at the right, they are $1, 5 ^{1} = 5, 5 ^{2} = 25, 5 ^{3} = 125,5 ^{4} = 125$

The lowest base five number with four digits would be

$1000 _{\textrm{five}} = 1 \times 5^{3} = 125$ in base ten.

The lowest base five number with five digits would be

$10000 _{\textrm{five}} = 1 \times 5^{4} = 625$ in base ten.

Therefore, a number that is expressed as a four-digit number in base five must fall in the range

$125 \leq N \leq 624$

Three of the four numbers - all except 100 - fall in this range.

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Example Question #13 : How To Add Exponents

Solve for $x$ :

$4^{4}+4^{4}+4^{4}+4^{4}=2^{x}$

Possible Answers:

$9$

$11$

$10$

$6$

$8$

Correct answer:

$10$

Explanation:

Combining the powers, we get $1024=2^{x}$ .

From here we can use logarithms, or simply guess and check to get $x=10$ .

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Example Question #14 : How To Add Exponents

If $\dpi{100} \small r$ and $\dpi{100} \small s$ are positive integers, and $\dpi{100} \small 25\left ( 5^{r} \right )=5^{s-2}$ , then what is $\dpi{100} \small s$ in terms of $\dpi{100} \small r$ ?

Possible Answers:

$\dpi{100} \small r+3$

$\dpi{100} \small r$

$\dpi{100} \small r+1$

$\dpi{100} \small r+2$

$\dpi{100} \small r+4$

Correct answer:

$\dpi{100} \small r+4$

Explanation:

$\dpi{100} \small 25\left ( 5^{r} \right )$ is equal to $5^{2}\left ( 5^{r}\right )$ which is equal to $\dpi{100} \small \left ( 5^{r+2} \right )$ . If we compare this to the original equation we get $\dpi{100} \small r+2=s-2\rightarrow s=r+4$

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Example Question #1 : How To Subtract Exponents

If m and n are integers such that m < n < 0 and m² – n² = 7, which of the following can be the value of m + n?

I. –5

II. –7

III. –9

Possible Answers:

II and III only

II only

I and II only

I, II and III only

I only

Correct answer:

II only

Explanation:

m and n are both less than zero and thus negative integers, giving us m² and n² as perfect squares. The only perfect squares with a difference of 7 is 16 – 9, therefore m = –4 and n = –3.

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Example Question #2 : How To Subtract Exponents

$\frac{7^5+7^4}{8}=$

Possible Answers:

$\frac{7}{8}$

$\frac{7^4}{8}$

7^4

$\frac{14^4}{8}$

$\frac{7^9}{8}$

Correct answer:

7^4

Explanation:

To simplify, we can rewrite the numerator using a common exponential base.

$\frac{7^5+7^4}{8}=\frac{7(7^4)+7^4}{8}$

Now, we can factor out the numerator.

$\frac{7(7^4)+7^4}{8}=\frac{7^4(7+1)}{8}=\frac{7^4(8)}{8}$

The eights cancel to give us our final answer.

$\frac{7^4(8)}{8}=7^4$

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Example Question #1 : How To Subtract Exponents

If $x\star y=x^2-y^2$ , then what is $4\star (5\star 3)$ ?

Possible Answers:

240

Correct answer:

Explanation:

$4\star (5\star 3)$

Follow the order of operations by solving the expression within the parentheses first.

$x\star y=x^2-y^2$

$5\star 3=(5)^2-(3)^2$

$5\star 3=25-9=16$

Return to solve the original expression.

$4\star(5\star3)=4\star(16)$

$x\star y=x^2-y^2$

$4\star16=(4)^2-(16)^2$

$4\star(16)=-240$

$4\star(5\star3)=-240$

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Example Question #3 : How To Subtract Exponents

Solve:

$4^{8}\div2^{10}=$

Possible Answers:

-0.5

Correct answer:

Explanation:

$4^{8}=\left ( 2^{2}\right )^{8}=2^{16}$

Subtract the denominator exponent from the numerator's exponent, since they have the same base.

$\frac{2^{16}}{2^{10}}=2^{16-10}=2^{6}=64$

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Example Question #12 : Exponents

5⁴/ 25 =

Possible Answers:

5⁴ / 5

Correct answer:

Explanation:

25 = 5 * 5 = 5². Then 5⁴ / 25 = 5⁴ / 5².

Now we can subtract the exponents because the operation is division. 5⁴ / 5²= 5^{4 – 2} = 5² = 25. The answer is therefore 25.

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PSAT Math : Exponential Operations

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #3 : How To Add Exponents

Example Question #81 : Exponential Operations

Example Question #13 : How To Add Exponents

Example Question #13 : How To Add Exponents

Example Question #14 : How To Add Exponents

Example Question #1 : How To Subtract Exponents

Example Question #2 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #3 : How To Subtract Exponents

Example Question #12 : Exponents

All PSAT Math Resources

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