SAT Math : SAT Mathematics

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #91 : Exponents

Evaluate:

\displaystyle 2^8+2^8

Possible Answers:

\displaystyle 2^{16}

\displaystyle 2^{15}

\displaystyle 4^8

\displaystyle 2^7

\displaystyle 2^9

Correct answer:

\displaystyle 2^9

Explanation:

When adding exponents, you want to factor out to make solving the question easier.

\displaystyle 2^8+2^8 we can factor out \displaystyle 2^8 to get 

\displaystyle 2^8(1+1)=2^8(2).

We have the same base so we just apply the exponent rule for multiplication to get 

\displaystyle 2^8(2)=2^{8+1}=2^9.

Example Question #92 : Exponents

Which of the following is equivalent to \displaystyle 16^6+4^{12}+64^{4}+256^3?

Possible Answers:

\displaystyle 64^3

\displaystyle 4^{13}

\displaystyle 16^8

\displaystyle 256^4

\displaystyle 4^{14}

Correct answer:

\displaystyle 4^{13}

Explanation:

Although each base is different, we can convert them to a common base of \displaystyle 4. 

We know 

\displaystyle 16^6=(4^2)^6=4^{12}

\displaystyle 64^4=(4^3)^4=4^{12},

and 

\displaystyle 256^3=(4^4)^3=4^{12}.

Remember to apply the power rule of exponents.

Therefore we have 

\displaystyle 4^{12}+4^{12}+4^{12}+4^{12}.

We can factor out \displaystyle 4^{12} to get 

\displaystyle 4^{12}(1+1+1+1)=4^{12}(4)=4^{13}.

Example Question #93 : Exponents

Simplify: \displaystyle 3^{2x}+3^{2x}+3^{2x}

 

Possible Answers:

\displaystyle 3^{2x}

\displaystyle 3^{6x}

\displaystyle 3^{3x}

\displaystyle 3^{2x+1}

\displaystyle 3^{3x+1}

Correct answer:

\displaystyle 3^{2x+1}

Explanation:

When adding exponents, you want to factor out to make solving the question easier.

\displaystyle 3^{2x}+3^{2x}+3^{2x} 

We can factor out \displaystyle 3^{2x} to get 

\displaystyle 3^{2x}(1+1+1)=3^{2x}*3.

Now we can add exponents and therefore our answer is 

\displaystyle 3^{2x+1}.

Example Question #503 : Algebra

Given  \displaystyle 9 \cdot 3^n = 3^9, what is the value of \displaystyle n?

Possible Answers:

5

9

11

7

3

Correct answer:

7

Explanation:

Express \displaystyle 9 as a power of \displaystyle 3; that is: \displaystyle 9=3^2.

Then \displaystyle 3^2 \cdot 3^n = 3^9.

Using the properties of exponents, \displaystyle 3^{n+2}=3^9.

Therefore, \displaystyle n+2=9, so \displaystyle n=7.

Example Question #1 : How To Add Exponents

If \displaystyle 9^{(x + 5)}+3^{2(x+5)}=162, what is the value of \displaystyle x?

Possible Answers:

\displaystyle 1

\displaystyle 0

\displaystyle -3

\displaystyle -1

Correct answer:

\displaystyle -3

Explanation:

Since we have two \displaystyle x’s in \displaystyle 9^{(x + 5)} + 3^{2(x + 5)} we will need to combine the two terms.

For \displaystyle 3^{2(x + 5) } this can be rewritten as

\displaystyle (3^2)^{ (x + 5)} = 9^ {(x + 5)}

So we have \displaystyle 9^{ (x + 5) }+ 9^{ (x + 5)} = 162.

Or \displaystyle 2 (9^{ (x + 5)}) = 162

Divide this by \displaystyle 2\displaystyle 9^{ (x + 5)} = 81 = 9^ 2

Thus \displaystyle x +5 = 2 or \displaystyle x = -3

*Hint: If you are really unsure, you could have plugged in the numbers and found that the first choice worked in the equation.

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:

I, II and III only

I and II only

II and III only

I only

II 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.

Example Question #2 : How To Subtract Exponents

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

Possible Answers:

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

\displaystyle \frac{7}{8}

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

\displaystyle 7^4

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

Correct answer:

\displaystyle 7^4

Explanation:

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

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

Now, we can factor out the numerator.

\displaystyle \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.

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

 

Example Question #3 : How To Subtract Exponents

Simplify the following expression:

\displaystyle \frac{x^{7}y^{3}}{x^{3}}

Possible Answers:

\displaystyle x^{4}y^{3}

\displaystyle x^{7}y^2

\displaystyle \frac{xy^{10}}{x^{3}}

\displaystyle xy

Correct answer:

\displaystyle x^{4}y^{3}

Explanation:

The correct answer can be found by subtracting exponents that have the same base. Whenever exponents with the same base are divided, you can subtract the exponent of the denominator from the exponent of the numerator as shown below to obtain the final answer:

\displaystyle \frac{x^{7}y^{3}}{x^{3}} = x^{7-3}y^{3} = x^{4}y^{3}

You do not do anything with the y exponent because it has no identical bases.

Example Question #1 : How To Subtract Exponents

Solve:

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

Possible Answers:

\displaystyle -1

\displaystyle 2

\displaystyle -0.5

\displaystyle 64

\displaystyle 16

Correct answer:

\displaystyle 64

Explanation:

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

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

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

Example Question #1 : How To Subtract Exponents

Simplify: 32 * (423 - 421)

Possible Answers:

None of the other answers

3^3 * 4^21 * 5

4^4

3^21

3^3 * 4^21

Correct answer:

3^3 * 4^21 * 5

Explanation:

Begin by noting that the group (423 - 421) has a common factor, namely 421.  You can treat this like any other constant or variable and factor it out.  That would give you: 421(42 - 1). Therefore, we know that:

32 * (423 - 421) = 32 * 421(42 - 1)

Now, 42 - 1 = 16 - 1 = 15 = 5 * 3.  Replace that in the original:

32 * 421(42 - 1) = 32 * 421(3 * 5)

Combining multiples withe same base, you get:

33 * 421 * 5

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