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Common Core: 8th Grade Math : Expressions & Equations

Study concepts, example questions & explanations for Common Core: 8th Grade Math

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All Common Core: 8th Grade Math Resources

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Example Questions

Example Question #1 : Simplifying Exponents

Simplify the expression:

$(3x^4y^2)(4xy^2)^{-3}$

Possible Answers:

$\frac{3x}{64y^3}$

$\frac{3x}{64y^4}$

Cannot be simplified

$\frac{3x}{16y^3}$

Correct answer:

$\frac{3x}{64y^4}$

Explanation:

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

$(3x^{4}y^2)(4xy^2)^{-3}= (3xy^2)(4^{-3}x^{-3}y^{-6})$

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

$\frac{3x^4y^2}{64x^3y^6}$

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

$\frac{3x^4y^2}{64x^3y^6}=\frac{3}{64}x^{4-3}y^{2-6}=\frac{3}{64}xy^{-4}$

$\frac{3x}{64y^4}$

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Example Question #1 : Distributing Exponents (Power Rule)

Simplify:

(4x^3)^4

Possible Answers:

4x^7

$256x^{12}$

512x

16x^7

$4x^{12}$

Correct answer:

$256x^{12}$

Explanation:

Use the power rule to distribute the exponent:

$(4x^3)^4\rightarrow 4^4\cdot x^{3\cdot 4}\rightarrow 256x^{12}$

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Example Question #11 : Distributing Exponents (Power Rule)

Simplify: (2^3)^2-(2^2+ 2^3)^2

Possible Answers:

Correct answer:

Explanation:

To simplify this, we will need to use the power rule and order of operations.

Evaluate the first term. This will be done in two ways to show that the power rule will work for exponents outside of the parenthesis for a single term.

$(2^3)^2 = 2^3\times 2^3 =8\times 8 = 64$

$(2^3)^2 = 2^{3\times 2} = 2^6 =64$

For the second term, we cannot distribute 2^2 and 2^3 with the exponent outside the parentheses because it's not a single term. Instead, we must evaluate the terms inside the parentheses first.

Evaluate the second term.

(2^2+ 2^3)^2 = (4+8)^2 = (12)^2

Square the value inside the parentheses.

$(12)^2 = 12\times 12 = 144$

Subtract the value of the second term with the first term.

64-144 = -80

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Example Question #1 : Expressions & Equations

Which of the following is equivalent to the expression (7x^5)^3 ?

Possible Answers:

$343x^{8}$

$49x^{2}$

$7x^{15}$

$343x^{15}$

Correct answer:

$343x^{15}$

Explanation:

Which of the following is equivalent to the expression (7x^5)^3 ?

We can rewrite the given expression by distributing the exponent on the outside.

(7x^5)^3=7^3(x^5)^3

Now, this may look a little messier, but we need to recall that when we distribute an exponent through parentheses as we are trying to do above, we need to multiple the exponent on the inside by the number on the outside.

In a general sense it looks like this:

$(a^b)^c=a^{b*c}$

For our specific problem, it looks like this:

$7^3(x^5)^3=7^3x^{15}=343x^{15}$

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Example Question #2 : Generate Equivalent Numerical Expressions: Ccss.Math.Content.8.Ee.A.1

Simplify.

$(x^4)^{6}$

Possible Answers:

x^2

$x^{10}$

$x^{-2}$

$x^{46}$

$x^{24}$

Correct answer:

$x^{24}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(x^4)^{6}=x^{4*6}=x^{24}$

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Example Question #14 : Distributing Exponents (Power Rule)

Simplify.

$(x^{-3})^{-4}$

Possible Answers:

$x^{12}$

$x^{-7}$

$x^{7}$

$x^{34}$

$x^{-12}$

Correct answer:

$x^{12}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(x^{-3})^{-4}=x^{-3*-4}=x^{12}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify.

$(\frac{x^5}{y^3})^3$

Possible Answers:

$\frac{x^{8}}{y^6}$

$\frac{x^{15}}{y^{13}}$

$\frac{x^{15}}{y^8}$

$\frac{x^{15}}{y^3}$

$\frac{x^{15}}{y^9}$

Correct answer:

$\frac{x^{15}}{y^9}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(\frac{x^5}{y^3})^{3}=\frac{x^{5*3}}{y^{3*3}}=\frac{x^{15}}{y^{9}}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify the expression $\left ( 2x^3\right )^5$

Possible Answers:

$2x^{15}$

$32768x^{15}$

32x^8

$32x^{15}$

2x^8

Correct answer:

$32x^{15}$

Explanation:

$\left ( 2x^5\right )^3=\left ( 2x^5\right )\left ( 2x^5\right )\left ( 2x^5\right )$

$8x^{15}$

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Example Question #1 : Generate Equivalent Numerical Expressions: Ccss.Math.Content.8.Ee.A.1

Evaluate $\dpi{100} \frac{2^{10}}{2^{8}}$

Possible Answers:

$\dpi{100} 2$

$\dpi{100} \frac{8}{5}$

$\dpi{100} 200$

$\dpi{100} 4$

Correct answer:

$\dpi{100} 4$

Explanation:

If you divide two exponential expressions with the same base, you can simply subtract the exponents. Here, both the top and the bottom have a base of 2 raised to a power.

So $\dpi{100} \frac{2^{10}}{2^{8}}=2^{10-8}=2^{2}=4$

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Example Question #1 : Properties Of Exponents

$\dpi{100} 2^{3}\cdot 2^{2}$

Possible Answers:

$\dpi{100} 16$

$\dpi{100} 2^{6}$

$\dpi{100} 32$

Correct answer:

$\dpi{100} 32$

Explanation:

Since the two expressions have the same base, we just add the exponents.

$\dpi{100} 2^{3}\cdot 2^{2}=2^{3+2}=2^{5}=32$

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Common Core: 8th Grade Math : Expressions & Equations

Study concepts, example questions & explanations for Common Core: 8th Grade Math

All Common Core: 8th Grade Math Resources

Example Questions

Example Question #1 : Simplifying Exponents

Example Question #1 : Distributing Exponents (Power Rule)

Example Question #11 : Distributing Exponents (Power Rule)

Example Question #1 : Expressions & Equations

Example Question #2 : Generate Equivalent Numerical Expressions: Ccss.Math.Content.8.Ee.A.1

Example Question #14 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #1 : Generate Equivalent Numerical Expressions: Ccss.Math.Content.8.Ee.A.1

Example Question #1 : Properties Of Exponents

All Common Core: 8th Grade Math Resources

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