Expressions & Equations - Common Core: 8th Grade Math

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Question

$\small 76a^3+19a^2+16a=-4$

Which of the following displays the full real-number solution set for in the equation above?

Answer

Rewriting the equation as $\small 76a^3+19a^2+16a+4=0$ , we can see there are four terms we are working with, so factor by grouping is an appropriate method. Between the first two terms, the Greatest Common Factor (GCF) is $\small 19a^2$ and between the third and fourth terms, the GCF is 4. Thus, we obtain $\small \small 19a^2(4a+1)+4(4a+1)=0 \rightarrow (4a+1)(19a^2+4)=0$ . Setting each factor equal to zero, and solving for $\small a$ , we obtain $\small a=-\frac{1}{4}$ from the first factor and $\small a^2=-\frac{4}{19}$ from the second factor. Since the square of any real number cannot be negative, we will disregard the second solution and only accept $\small a=-\frac{1}{4}$ .

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Question

Solve for $x\textup:$

Answer

We can solve this problem one of two ways: first we can ask ourselves the question:

"What number cubed is equal to "

If you aren't sure of the answer to this question, then you can solve the problem algebraically.

In order to solve this problem using algebra, we need to isolate the on one side of the equation. Remember that operations done to one side of the equation must be performed on the opposite side.

We will solve this equation by performing the opposite operation of cubing a number, which is taking the cubed root:

$\sqrt[3]{x}=\sqrt[3]{8}$

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Question

Solve for $x\textup:$

Answer

We can solve this problem one of two ways: first we can ask ourselves the question:

"What number cubed is equal to "

If you aren't sure of the answer to this question, then you can solve the problem algebraically.

In order to solve this problem using algebra, we need to isolate the on one side of the equation. Remember that operations done to one side of the equation must be performed on the opposite side.

We will solve this equation by performing the opposite operation of cubing a number, which is taking the cubed root:

$\sqrt[3]{x}=\sqrt[3]{64}$

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Question

Solve for $x\textup:$

Answer

We can solve this problem one of two ways: first we can ask ourselves the question:

"What number cubed is equal to "

If you aren't sure of the answer to this question, then you can solve the problem algebraically.

In order to solve this problem using algebra, we need to isolate the on one side of the equation. Remember that operations done to one side of the equation must be performed on the opposite side.

We will solve this equation by performing the opposite operation of cubing a number, which is taking the cubed root:

$\sqrt[3]{x}=\sqrt[3]{27}$

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Question

Solve for $x\textup:$

Answer

We can solve this problem one of two ways: first we can ask ourselves the question:

"What number cubed is equal to "

If you aren't sure of the answer to this question, then you can solve the problem algebraically.

In order to solve this problem using algebra, we need to isolate the on one side of the equation. Remember that operations done to one side of the equation must be performed on the opposite side.

We will solve this equation by performing the opposite operation of cubing a number, which is taking the cube root:

$\sqrt[3]{x}=\sqrt[3]{125}$

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Question

Solve for $x\textup:$

Answer

We can solve this problem one of two ways: first we can ask ourselves the question:

"What number cubed is equal to "

If you aren't sure of the answer to this question, then you can solve the problem algebraically.

In order to solve this problem using algebra, we need to isolate the on one side of the equation. Remember that operations done to one side of the equation must be performed on the opposite side.

We will solve this equation by performing the opposite operation of cubing a number, which is taking the cubed root:

$\sqrt[3]{x}=\sqrt[3]{216}$

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Question

Solve for $x\textup:$

Answer

We can solve this problem one of two ways: first we can ask ourselves the question:

"What number cubed is equal to "

If you aren't sure of the answer to this question, then you can solve the problem algebraically.

In order to solve this problem using algebra, we need to isolate the on one side of the equation. Remember that operations done to one side of the equation must be performed on the opposite side.

We will solve this equation by performing the opposite operation of cubing a number, which is taking the cube root:

$\sqrt[3]{x}=\sqrt[3]{343}$

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Question

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

Answer

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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Question

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

Answer

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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Question

Evaluate: $\dpi{100} (3^{3})^{2}$

Answer

A power raised to a power indicates that you multiply the two powers.

$\dpi{100} (3^{3})^{2}=3^{3\cdot 2}=3^{6}$

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Question

Simplify the expression:

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

Answer

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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Question

Simplify:

$\frac{8x ^{-2}}{(3x)^{2}}$

Answer

$\frac{8x ^{-2}}{(3x)^{2}}$

To solve this problem, we start with the parentheses and exponents in the denominator.

$= \frac{8x ^{-2}}{3^{2}x^{2}} = \frac{8x ^{-2}}{9x^{2}}$

Next, we can bring the from the denominator up to the numerator by making the exponent negative.

$= \frac{8x ^{-2-2}}{9} = \frac{8x ^{-4}}{9}$

Finally, to get rid of the negative exponent we can bring it back down to the denominator.

$= \frac{8}{9x ^{4}}$

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Question

Evaluate:

$\frac{4^{4}}{4^{4} +\left ( -4 \right ) ^{4} }$

Answer

$4^{4} = 4 \times 4 \times 4 \times 4 = 256$

$\left (- 4 \right ) ^{4} = \left (- 4 \right ) \times \left (- 4 \right ) \times \left (- 4 \right ) \times \left (- 4 \right ) = 256$

$\frac{4^{4}}{4^{4} +\left ( -4 \right ) ^{4} } = \frac{256 }{256 + 256} = \frac{256 }{512} =\frac{1 }{2}$

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Question

Evaluate:

$(100 - 4 \cdot 25) ^{100 \div 25 }$

Answer

$(100 - 4 \cdot 25) ^{100 \div 25 } = (100 - 100) ^{4} = 0 ^{4} =0$

as 0 taken to any positive power is equal to 0.

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Question

Evaluate:

$(80 - 4 \cdot 25) ^{100 \div 25 }$

Answer

$(80 - 4 \cdot 25) ^{100 \div 25 }$

$= (80 - 100) ^{4 }$

$= (-20) ^{4 }$

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Question

Simplify:

Answer

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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Question

Evaluate:

$3^{-2}\times3^{-6}\times3^{6}$

Answer

The bases of all three terms are alike. Since the terms are of a specific power, the rule of exponents state that the powers can be added if the terms are multiplied.

$3^{-2}\times3^{-6}\times3^{6}= 3^{-2-6+6}= 3^{-2}$

When we have a negative exponent, we we put the number and the exponent as the denominator, over

$\frac{1}{3^2}=\frac{1}{9}$

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Question

Simplify:

Answer

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 and 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.

Square the value inside the parentheses.

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

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

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Question

Which of the following is equivalent to the expression ?

Answer

Which of the following is equivalent to the expression ?

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

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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Question

Simplify.

$(x^4)^{6}$

Answer

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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