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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

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16 Diagnostic Tests 660 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #61 : Exponents

Simplify:

$4^7*4^{18}$

Possible Answers:

$4^{25}$

$4^{45}$

$4^{106}$

$4^{35}$

$4^{126}$

Correct answer:

$4^{25}$

Explanation:

When multiplying exponents, we just add the exponents while keeping the base the same.

$4^7*4^{18}=4^{7+18}=4^{25}$

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Example Question #62 : Exponents

Simplify:

7^8*5^8

Possible Answers:

12^8

35^8

$35^{16}$

$12^{16}$

57^8

Correct answer:

35^8

Explanation:

When multiplying exponents with different bases but the same exponent, you multiply the bases and keep the exponents the same.

7^8*5^8=(7*5)^8=35^8

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Example Question #63 : Exponents

Define an operation $\bigstar$ on the set of real numbers as follows:

For all real a,b ,

$a \bigstar b = a ^{2+ b }$

Evaluate $100 \bigstar (-2)$

Possible Answers:

$\frac{1}{100}$

100

The value of $100 \bigstar (-2)$ is an undefined quantity.

Correct answer:

Explanation:

Set a =100 and b= 2 in the expression in the definition, then simplify the exponent:

$a \bigstar b = a ^{2+ b }$

$100 \bigstar (-2)= 100 ^{2+ (-2)}$

$100 \bigstar (-2)= 100 ^{0}$

Any nonzero number raised to the power of 0 is equal to 1, so

$100 \bigstar (-2)= 1$ .

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Example Question #63 : Exponential Operations

Define an operation $\bigstar$ on the set of real numbers as follows:

For all real a,b ,

$a \bigstar b =\left ( \frac{1}{a} \right )^ {\frac{1}{b}}$

Evaluate: $100 \bigstar \frac{1}{2}$

Possible Answers:

$\frac{1}{10,000}$

$\frac{1}{10}$

10,000

-10,000

Correct answer:

$\frac{1}{10,000}$

Explanation:

Set a =100 and $b= \frac{1}{2}$ in the expression in the definition:

$a \bigstar b =\left ( \frac{1}{a} \right )^ {\frac{1}{b}}$

$100 \bigstar \frac{1}{2 }=\left ( \frac{1}{100} \right )^ {\frac{1}{ \frac{1}{2}}}$

The exponent $\frac{1}{ \frac{1}{2}}$ simplifies as follows:

$\frac{1}{ \frac{1}{2}} = 1 \div \frac{1}{2} = 1 \cdot 2 = 2$ , so

$100 \bigstar \frac{1}{2 }=\left ( \frac{1}{100} \right )^ {2}= \frac{1}{100} \cdot \frac{1}{100} = \frac{1}{10,000}$

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Example Question #62 : Exponents

Convert the product of 16^8*2^4 to base .

Possible Answers:

$2^{12}$

$2^{36}$

$2^{64}$

$2^{24}$

$2^{16}$

Correct answer:

$2^{36}$

Explanation:

Although they have different bases, we know that 16=2^4 .

Therefore

$16^8=(2^4)^8=2^{32}$ .

Remember to apply the power rule of exponents.

Finally,

$2^{32}*2^4=2^{32+4}=2^{36}$ .

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Example Question #61 : How To Multiply Exponents

Solve for :

$\sqrt[3]{x-3} = 2$

Possible Answers:

$None\ of\ the\ above$

Correct answer:

Explanation:

$\left ( \sqrt[3]{x-3} \right )^{3} = 2^{3}$

Now the left side equals x-3 and the right side equals 8. Hence:

x-3 = 8

Therefore must be equal to 11.

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Example Question #63 : Exponents

Define an operation $\bigstar$ on the set of real numbers as follows:

For all real a,b ,

$a \bigstar b =\left ( \frac{1}{a} \right )^ {\frac{1}{b}}$

Evaluate: $81 \bigstar (-2)$ .

Possible Answers:

$\frac{1}{9}$

$\frac{1}{6,561}$

6,561

The value of $81 \bigstar (-2)$ is undefined.

Correct answer:

Explanation:

Set a =81 and b= -2 in the expression in the definition, then simplify:

$a \bigstar b =\left ( \frac{1}{a} \right )^ {\frac{1}{b}}$

$81\bigstar (-2) =\left ( \frac{1}{81} \right )^ {\frac{1}{-2}} = \left ( \frac{1}{81} \right )^ {-\frac{1}{2}}$

By definition,

$a ^{-M} = \frac{1}{a^{M}} = \frac{1^{M}}{a^{M}}= \left ( \frac{1}{a} \right )^{M}$ ;

setting N = -M ,

$\left ( \frac{1}{a} \right )^{-N} = a ^{N}$ .

Substituting a = 81 and $N = -\frac{1}{2}$ ,

$81\bigstar (-2) = \left ( \frac{1}{81} \right )^ {-\frac{1}{2}} = 81 ^ {\frac{1}{2}}$

Also, raising a number to the power of $\frac{1}{M}$ is equivalent to taking the th root of the number; specifically, raising a number to the power of $\frac{1}{2}$ is equivalent to taking its square root. Therefore,

$81\bigstar (-2) = 81 ^ {\frac{1}{2}} = \sqrt{81} = 9$ .

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Example Question #472 : Algebra

$\sqrt[6]{x} = t$

Express $\sqrt{x}$ in terms of .

Possible Answers:

$t ^{4}$

$t^{3}$

$t \sqrt{t}$

Correct answer:

$t^{3}$

Explanation:

By definition,

$\sqrt[6]{x} = x^{\frac{1}{6}}$

and

$\sqrt{x} = x^{\frac{1}{2}}$ .

By the Power of a Power Rule,

$x^{\frac{1}{2}} = x^{\frac{1}{6}\cdot 3 } = \left ( x ^{\frac{1}{6}}\right ) ^{3}$ ,

so, by substituting,

$\sqrt{x} = \left ( \sqrt[6]{x} \right ) ^{3} = t^{3}$ .

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Example Question #61 : Exponential Operations

Simplify the expression.

x^3(2x^2)

Possible Answers:

4x^5

2x^5

2x^7

4x^6

2x^6

Correct answer:

2x^5

Explanation:

When multiplying with exponents, you must add the exponents.

Therefore, multiply the coefficients on the x terms and add the exponents

x^3(2x^2)

$\\1x^3\cdot2x^2=2x^(3+2)=2x^5$

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Example Question #62 : Exponential Operations

Which of the following is equivalent to (3x^4)(2x^2)

Possible Answers:

6x^8

5x^6

5x^8

6x^6

Correct answer:

6x^6

Explanation:

and can be multiplied together to give you which is the first part of our answer. When you multiply exponents with the same base (in this case, ), you add the exponents. In this case, $x^4\times x^2$ should give us x^6 because 4+2=6 . The answer is 6x^6

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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #61 : Exponents

Example Question #62 : Exponents

Example Question #63 : Exponents

Example Question #63 : Exponential Operations

Example Question #62 : Exponents

Example Question #61 : How To Multiply Exponents

Example Question #63 : Exponents

Example Question #472 : Algebra

Example Question #61 : Exponential Operations

Example Question #62 : Exponential Operations

All SAT Math Resources

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