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ACT Math : Algebra

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Example Questions

Example Question #21 : How To Multiply Exponents

The expression $a^{10}+b^{10}$ $\displaystyle a^{10}+b^{10}$ is equivalent to which of the following?

Possible Answers:

$\displaystyle ((a^5)^2)+((b^2)^5)$

$\frac{(a^{20}+b^{20})}{2}$ $\displaystyle \frac{(a^{20}+b^{20})}{2}$

$\displaystyle 2(a^5+b^5)$

None of these

$\displaystyle (a^2+b^2)^5$

Correct answer:

$\displaystyle ((a^5)^2)+((b^2)^5)$

Explanation:

The formula for multiplying exponents is

$a^m \cdot a^n=a^{m+n}$ $\displaystyle a^m \cdot a^n=a^{m+n}$.

Using this, we see that

$((a^5)^2) = a^5 \cdot a^5 = a^{10}$ $\displaystyle ((a^5)^2) = a^5 \cdot a^5 = a^{10}$, and

$(b^2)^5=b^2 \cdot b^2 \cdot b^2 \cdot b^2 \cdot b^2 = b^{10}$ $\displaystyle (b^2)^5=b^2 \cdot b^2 \cdot b^2 \cdot b^2 \cdot b^2 = b^{10}$.

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Example Question #721 : Algebra

Simplify the following: $x^{5}\cdot x^{5}$ $\displaystyle x^{5}\cdot x^{5}$

Possible Answers:

$x^{225}$ $\displaystyle x^{225}$

$x^{10}$ $\displaystyle x^{10}$

$x^{5}$ $\displaystyle x^{5}$

$x^{25}$ $\displaystyle x^{25}$

$x^{15}$ $\displaystyle x^{15}$

Correct answer:

$x^{10}$ $\displaystyle x^{10}$

Explanation:

When two variables with exponents are multiplied, you can simplify the expression by adding the exponents together. In this particular problem, the correct answer is found by adding the exponents 5 and 5, yielding $x^{10}$ $\displaystyle x^{10}$.

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Example Question #722 : Algebra

Simplify the following to its simplest exponential expression: $5^{2}\cdot5^{2}$ $\displaystyle 5^{2}\cdot5^{2}$

Possible Answers:

$\displaystyle 50$

$10^{2}$ $\displaystyle 10^{2}$

$25^{2}$ $\displaystyle 25^{2}$

$25^{4}$ $\displaystyle 25^{4}$

$5^{4}$ $\displaystyle 5^{4}$

Correct answer:

$5^{4}$ $\displaystyle 5^{4}$

Explanation:

When multiplying exponential expressions, the bases remain the same and the exponents are added. Thus, the answer to this question is $5^{4}$ $\displaystyle 5^{4}$.

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Example Question #723 : Algebra

$2^{6}$ $\displaystyle 2^{6}$ can be written as which of the following?

A. $(2^{3})^{2}$ $\displaystyle (2^{3})^{2}$

B. $\frac{1}{2^{-5}}$ $\displaystyle \frac{1}{2^{-5}}$

C. $\displaystyle 64$

Possible Answers:

C only

A, B and C

A only

B and C

A and C

Correct answer:

A and C

Explanation:

B is not equivalent because...

$\frac{1}{2^{-5}} = 2^{5} = 32$ $\displaystyle \frac{1}{2^{-5}} = 2^{5} = 32$

A is equivalent because of a property of exponents meaning that $(a^b)^c = a^{bc}$ $\displaystyle (a^b)^c = a^{bc}$. Consequently,

$(2^{3})^{2} = 2^{6}$ $\displaystyle (2^{3})^{2} = 2^{6}$

C is simply computing $2^{6} = 32$ $\displaystyle 2^{6} = 32$.

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Example Question #21 : How To Multiply Exponents

Solve $3x^{3}+2y$ $\displaystyle 3x^{3}+2y$ when x=2 $\displaystyle x=2$ and y=4 $\displaystyle y=4$.

Possible Answers:

$\displaystyle 30$

$\displaystyle 26$

$\displaystyle 36$

$\displaystyle 32$

Correct answer:

$\displaystyle 32$

Explanation:

Substitute $\displaystyle 2$ for $\displaystyle x$ and $\displaystyle 4$ for $\displaystyle y$: $3(2^{3})+2(4)$ $\displaystyle 3(2^{3})+2(4)$

Simplify: 3(8)+8 $\displaystyle 3(8)+8$

24+8=32 $\displaystyle 24+8=32$

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Example Question #724 : Algebra

What is $(4^{3})^{2}$ $\displaystyle (4^{3})^{2}$?

Possible Answers:

1600 $\displaystyle 1600$

4096 $\displaystyle 4096$

1024 $\displaystyle 1024$

1000 $\displaystyle 1000$

Correct answer:

4096 $\displaystyle 4096$

Explanation:

When an exponent is raised to an exponent, you may simplify by multiplying the exponents together to make a new exponent. In this case, $(4^{3})^{2}$ $\displaystyle (4^{3})^{2}$ becomes $4^{6}$ $\displaystyle 4^{6}$, which equals 4096 $\displaystyle 4096$.

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Example Question #725 : Algebra

Simplify: $x^{6}(x^{4})$ $\displaystyle x^{6}(x^{4})$

Possible Answers:

$x^{24}$ $\displaystyle x^{24}$

$x^{10}$ $\displaystyle x^{10}$

$x^{2}$ $\displaystyle x^{2}$

$x^{32}$ $\displaystyle x^{32}$

Correct answer:

$x^{10}$ $\displaystyle x^{10}$

Explanation:

When exponents with the same base are being multiplied, you may add the exponents together to create a new exponent.

In this case, you would add 6 and 4 to create 10 as the new exponent.

Keeping the same base, the answer becomes $x^{10}$ $\displaystyle x^{10}$.

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Example Question #21 : How To Multiply Exponents

Solve: $3x^{4}-2y^{2}$ $\displaystyle 3x^{4}-2y^{2}$ when x=2 $\displaystyle x=2$ and y=1 $\displaystyle y=1$.

Possible Answers:

$\displaystyle 48$

$\displaystyle 46$

$\displaystyle 44$

$\displaystyle 45$

Correct answer:

$\displaystyle 46$

Explanation:

Substitute $\displaystyle 2$ for $\displaystyle x$ and $\displaystyle 1$ for $\displaystyle y$: $3(2^{4})-2(1^{2})$ $\displaystyle 3(2^{4})-2(1^{2})$.

Simplify: 3(16)-2 $\displaystyle 3(16)-2$

48-2=46 $\displaystyle 48-2=46$

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Example Question #31 : Exponential Operations

Simplify: $(y^{3})^{4}$ $\displaystyle (y^{3})^{4}$

Possible Answers:

$y^{12}$ $\displaystyle y^{12}$

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

$y^{\frac{3}{4}}$ $\displaystyle y^{\frac{3}{4}}$

$y^{7}$ $\displaystyle y^{7}$

Correct answer:

$y^{12}$ $\displaystyle y^{12}$

Explanation:

When an exponent is raised to an exponent, multiply the two together to yield a new exponent, and attach that exponent to the original base.

In this case, multiply $\displaystyle 4$ and $\displaystyle 3$ to yield $\displaystyle 12$, and your answer is $y^{12}$ $\displaystyle y^{12}$.

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Example Question #32 : How To Multiply Exponents

Solve: $(x^{2})^{2}$ $\displaystyle (x^{2})^{2}$ when x=4 $\displaystyle x=4$.

Possible Answers:

256 $\displaystyle 256$

$\displaystyle 4$

$\displaystyle 16$

$\displaystyle 0$

Correct answer:

256 $\displaystyle 256$

Explanation:

When an exponent is raised to another exponent, you may multiply them together to find the exponent of the answer. Attach that exponent to the original base, and that is the solution.

In this particular problem, multiply $\displaystyle 2$ by $\displaystyle 2$, which yields $\displaystyle 4$ as the final exponent. This makes $x^{4}$ $\displaystyle x^{4}$.

Now, substitute $\displaystyle 4$ for $\displaystyle x$: $4^{4}=256$ $\displaystyle 4^{4}=256$.

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #21 : How To Multiply Exponents

Example Question #721 : Algebra

Example Question #722 : Algebra

Example Question #723 : Algebra

Example Question #21 : How To Multiply Exponents

Example Question #724 : Algebra

Example Question #725 : Algebra

Example Question #21 : How To Multiply Exponents

Example Question #31 : Exponential Operations

Example Question #32 : How To Multiply Exponents

All ACT Math Resources

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