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Pre-Algebra : Polynomials

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Example Questions

Example Question #71 : Polynomials

Simplify: $\left (x^{-2} \right )^{-3}$

Possible Answers:

$x^{-6}$

$x^{-1}$

$x^{-5}$

x^6

$x^{\frac{2}{3}}$

Correct answer:

x^6

Explanation:

The general rule for taking a power to a power is just that you multiply the exponents. So for $\left (x^{-2} \right )^{-3}$ you'd just multiply -2*-3=6 .

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Example Question #72 : Polynomials

Simplify the following expression:

$(2a^{3}b^{5}c^{2})^{3}$

Possible Answers:

$=2a^{9}b^{15}c^{6}$

$=8a^{1}b^{2}c^{-1}$

$=8a^{6}b^{8}c^{5}$

$=8a^{9}b^{15}c^{6}$

$=2a^{3}b^{3}c^{3}$

Correct answer:

$=8a^{9}b^{15}c^{6}$

Explanation:

When there is a product inside parenthesis and everything in the parenthesis is raised to a power, each term inside the parenthesis is taken to that power. If the terms inside the parenthesis are already raised to a power, multiply the exponents.

$(2a^{3}b^{5}c^{2})^{3}$

$=2^{3}a^{(3*3)}b^{(5*3)}c^{(2*3)}$

$=8a^{9}b^{15}c^{6}$

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Example Question #73 : Polynomials

Combine the expression into the least amount of bases and powers:

$([(2x^2y)^{2}]^4)^{-1}$

Possible Answers:

$\frac{1}{256x^{16}y^{16}}$

$\frac{1}{2x^{16}y^{8}}$

None of these answers.

$\frac{1}{4x^{16}y^{8}}$

$\frac{1}{256x^{16}y^{8}}$

Correct answer:

$\frac{1}{256x^{16}y^{8}}$

Explanation:

The power rule of exponents states that $(a^m)^n=a^{m*n}$ . In other words, one base "a" to two different powers is equal to "a" with its exponent being the product of those two powers.

Start from the middle and keep track of each power.

$([(2x^2y)^{2}]^4)^-1$

$((2^2x^{2*2}y^2)^4)^{-1}=((4x^4y^2)^4)^{-1}=((4^4x^{4*4}y^{2*4})^{-1})=(256x^{16}y^8)^{-1}$

Change to a positive exponent. $a^{-x}=\frac{1}{a^x}$ :

$(256x^{16}y^8)^{-1}=\frac{1}{(256x^{16}y^8)}$

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Example Question #15 : Power Rule Of Exponents

Simplify the following expression:

$\left ( x^{3}\right )^{4} + \left ( y^{7}\right )^{2} - \left ( x^{2}\right )^{0}$

Possible Answers:

$\left ( x^{12}\right ) + \left ( y^{14}\right )$

$\left ( x^{5}\right ) + \left ( y^{14}\right ) - 1$

$\left ( x^{12}\right ) + \left ( y^{14}\right ) - x$

$\left ( x^{12}\right ) + \left ( y^{14}\right ) - 1$

$\left ( x^{7}\right ) + \left ( y^{9}\right ) - 1$

Correct answer:

$\left ( x^{12}\right ) + \left ( y^{14}\right ) - 1$

Explanation:

First, the zero-exponent rule states that anything raised to the zero power is equal to :

$\left ( x^{3}\right )^{4} + \left ( y^{7}\right )^{2} - \left ( x^{2}\right )^{0}$

$\left ( x^{3}\right )^{4} + \left ( y^{7}\right )^{2} - 1$

Next, use the power rule (to raise a power to a power, multiply the exponents) to simplify the and variable-containing terms. This gives the expression in its simplest form:

$\left ( x^{12}\right ) + \left ( y^{14}\right ) - 1$

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Example Question #16 : Power Rule Of Exponents

Simplify this expression: (3x^2y^3)^3

Possible Answers:

3x^6y^9

27x^5y^6

The answer is not present.

27x^6y^9

3x^5y^6

Correct answer:

27x^6y^9

Explanation:

When an exponent is raised to another power or another exponent we multiply them together. This is called the power rule. $(a^m)^n=a^{m*n}$ .

(3x^2y^3)^3

$(3x^2y^3)^3=(3^3x^{2*3}y^{3*3})=27x^6y^9$

Everything within the parenthesis is raised to the power of three.

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Example Question #17 : Power Rule Of Exponents

Solve the equation.

4^3 * 4^7 =

Possible Answers:

4^2^1

4^-^4

4^4

16^1^0

4^1^0

Correct answer:

4^1^0

Explanation:

When solving and exponent question first look at the sign in the equation.

When it's multiplication you have to add the exponents.

The base number in this case is 4. If the base number is the same leave it be and take it with you to the answer, if it is different then you have to do what the sign says. So in this case if they were different bases you would have to multiply them, however they are the same so take the 4 to the answer and add the exponents.

4^3 ^+^7

4^1^0

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Example Question #18 : Power Rule Of Exponents

Simplify: (2x^3)^3

Possible Answers:

6x^9

8x^9

8x^6

6x^6

$8x^{27}$

Correct answer:

8x^9

Explanation:

One way to solve this question is to write out the terms in this expression.

$(2x^3)^3= 2x^3\cdot2x^3\cdot2x^3 = 8x^9$

With the exponent rule, notice that (2x^3)^3 can also be written as:

$(2x^3)^3= 2^3 \cdot x^{3\times3} = 8x^9$

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Example Question #19 : Power Rule Of Exponents

Solve: $6x^3 \times 4x^2$

Possible Answers:

24x^6

10x^6

$\textup{Cannot be multiplied since they are not like terms.}$

10x^5

24x^5

Correct answer:

24x^5

Explanation:

Multiply the coefficients together.

$6\times4 =24$

Multiply $x^3 \times x^2$ . When multiplying powers of the same base, their powers can be added.

$x^3 \times x^2= x^5$

The answer is: 24x^5

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Example Question #20 : Power Rule Of Exponents

Combine to one term: $a^4b^6 \times a^{-4} b^{-3}$

Possible Answers:

$\frac{1}{ab^2}$

$\frac{1}{a^{16}b^{18}}$

b^3

1+b^3

$\frac{b^2}{a}$

Correct answer:

b^3

Explanation:

Since both terms are multiplied, the powers of each base can be added or subtracted.

According to the rules of exponents,

$a^n\times a^m=a^{n+m}$

Therefore, our terms can combine as follows.

$a^4b^6 \times a^{-4} b^{-3} =a^{4-4} b^{6-3} = a^0b^3 = b^3$

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Example Question #74 : Polynomials

Simplify: $x^{-3}\times \frac{1}{x^3}\times x^{-3}$

Possible Answers:

$\frac{1}{x^3}$

$\frac{1}{x^6}$

x^3

x^6

$\frac{1}{x^9}$

Correct answer:

$\frac{1}{x^9}$

Explanation:

To simplify this, first convert the second term to a negative exponent.

$x^{-3}\times \frac{1}{x^3}\times x^{-3}= x^{-3}\times x^{-3}\times x^{-3}$

Since similar bases are multiplied, their powers can be added.

$x^{-3-3-3} = x^{-9} = \frac{1}{x^9}$

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Pre-Algebra : Polynomials

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

Example Questions

Example Question #71 : Polynomials

Example Question #72 : Polynomials

Example Question #73 : Polynomials

Example Question #15 : Power Rule Of Exponents

Example Question #16 : Power Rule Of Exponents

Example Question #17 : Power Rule Of Exponents

Example Question #18 : Power Rule Of Exponents

Example Question #19 : Power Rule Of Exponents

Example Question #20 : Power Rule Of Exponents

Example Question #74 : Polynomials

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