How to use the product rule for exponents in pre-algebra

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Math › How to use the product rule for exponents in pre-algebra

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Evaluate $\dpi{100} \frac{x^{(4x^{2}-3x+7)}}{x^{(2x^{2}+2x-6)}}$

$x^{2x^{2}-5x+13}$

$x^{4x-7}$

$\dpi{100} x^{4x^{2}-10x+26}$

$\dpi{100} x^{x^{2}-x+7}$

$\dpi{100} x^{x^{2}-2x+6}$

Explanation

The quotient rule for exponents states that when like numbers are raised to a certain power, then their exponents are subtracted from one another.

$\dpi{100} \dpi{100} \frac{x^{(4x^{2}-3x+7)}}{x^{(2x^{2}+2x-6)}}=x^{(4x^{2}-3x+7)-(2x^{2}+2x-6)}$

Solve for the exponent equation.

$(4x^{2}-3x+7)-(2x^{2}+2x-6)=4x^{2}-3x+7-2x^{2}-2x+6$

Combine like terms.

$\rightarrow (2x^{2}-5x+13)$

Therefore,

$\dpi{100} \dpi{100} \frac{x^{(4x^{2}-3x+7)}}{x^{(2x^{2}+2x-6)}}=x^{2x^{2}-5x+13}$

Which of the following is a simplified expression of X3X2?

$X^{5}$

$X^{6}$

$X\cdot 6$

$X^{1}$

$X\cdot X^{2}$

Explanation

When you multiply two exponential expressions with the same base, add the exponents to simplify the expression.

$X^{3}X^{2}=X^{5}$

Simplify the exponential expression.

$\small [(3x^2y^2)(4xy^3)]^0$

$\small 12x^3y^5$

undefined

Explanation

$\small [(3x^2y^2)(4xy^3)]^0$

Any number to the "zero" power is equal to .

$\small [(3x^2y^2)(4xy^3)]^0=[12x^3y^5]^0=1$

Evaluate:

$x^{(2x-2)}*x^{(4-2x)}$

$x^{2}$

$x^{2x}$

$x^{4-x}$

$x^{2x-4}$

$x^{4x-6}$

Explanation

The product rule for exponents states that when like numbers are raised to a certain power, their exponents are added to one another.

$x^{m}*x^{n}=x^{(m+n)}$

$x^{2x-2}*x^{4-2x}=x^{(2x-2)+(4-2x)}$

Solve for the exponent equation.

$\rightarrow 2$

Therefore,

$\dpi{100} x^{2x-2}*x^{4-2x}=x^{2}$ .

Which of the following is an alternate expression of X3Y3?

$(X\cdot Y)^{3}$

$(XY)^{6}$

$(X\cdot Y)^{9}$

$(X+Y)^{3}$

$X^{6}$

Explanation

When you are required to multiply exponential expressions with different bases but the same exponent, you can simplify the expression by adding parentheses and only using the exponent one time.

$X^{3}Y^{3}=(X\cdot Y)^{3}$

Simplify

$x^{11}$

$x^{24}$

$x^{13}$

$x^{14}$

Explanation

When multiplying the same variable raised to different powers, add the powers of each term together. In this case, add and together to find the power of the answer, .

Simplify $2^{4}2^{12}$ without evaluating.

$2^{16}$

$2^{24}$

$2^{8}$

$2^{14}$

Cannot be simplified

Explanation

When multilping 2 numbers with exponents, if the bases are the same, they can be combined and the exponents add together.

Simplify the expression.

Explanation

Begin by distributing the exponent into the parentheses. Remember that each term in the parentheses must be squared.

All terms are now being multiplied. To make it simpler, we can re-group by like terms.

Multiplying exponential terms with the same base allows us to add the exponents.

$12x^{2+2}y^{1+4}=12x^4y^5$

Simplify the expression.

Explanation

Since all terms are being multiplied, we can simplify by grouping like terms together.

To simplify the variable, we can add the exponents according to exponential identities.

$(2*5)(x^{2+3})$

Which of the following is equivalent to the expression:

$3^{6}\cdot3^{-4}$ ?

$\frac{1}{9}$

$\frac{1}{27}$

Explanation

Recall that when we are multiplying exponential expressions with the same base, we add the exponents and keep the base the same. Additionally, remember that adding a negative number is equivalent to subtracting.

$3^{6} \cdot3^{-4} = 3^{6 + -4} = 3^{2} = 9$

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