About Product & Power Rules for Exponents:
When we work exponents, we often need to use the product and power rules in order to simplify an expression. The product rule for exponents tells us that when we multiply two exponential expressions with the same base, we keep the base the same and add exponents. The power rules involve the power to power rule, the product to a power rule, and the quotient to a power rule. If we need to raise a power to another power, we keep the base the same and multiply the exponents together. Additionally, we will work with an exponent of zero. When a non-zero expression is raised to the power of zero, the result is always 1.
Test Objectives
- Demonstrate the ability to simplify using the product rule for exponents
- Demonstrate the ability to simplify using the power to power rule for exponents
- Demonstrate the ability to simplify by raising a product to a power
- Demonstrate the ability to simplify by raising a quotient to a power
- Demonstrate an understanding of an exponent of zero
#1:
Instructions: Simplify each.
$$a)\hspace{.2em}x^3 \cdot 2x^3$$
$$b)\hspace{.2em}x^3 \cdot x^2$$
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#2:
Instructions: Simplify each.
$$a)\hspace{.2em}3x^2 \cdot 2x^2$$
$$b)\hspace{.2em}{-}x^5y^4 \cdot -x^2y^2 \cdot (x^2y^2)^5$$
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#3:
Instructions: Simplify each.
$$a)\hspace{.2em}(-x)^5 \cdot x^5y^2$$
$$b)\hspace{.2em}(-x^2y^3 \cdot -x^2y^4 \cdot -x^3)^2$$
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#4:
Instructions: Simplify each.
$$a)\hspace{.2em}(-x^4)^5 \cdot -xy$$
$$b)\hspace{.2em}\left(\frac{x^2y^2 \cdot (xy)^5}{z^2 \cdot (q^5)^7}\right)^0$$
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#5:
Instructions: Simplify each.
$$a)\hspace{.2em}\left(\frac{-xy^4 \cdot (-x^2)^2}{z^2 \cdot (q^3)^0}\right)^2$$
$$b)\hspace{.2em}\frac{(2x^2 \cdot -2y^8)^2 \cdot (2x^5 \cdot y^3)^3}{(z^2q^0w^3)^3}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}2x^6$$
$$b)\hspace{.2em}x^5$$
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#2:
Solutions:
$$a)\hspace{.2em}6x^4$$
$$b)\hspace{.2em}x^{17}y^{16}$$
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#3:
Solutions:
$$a)\hspace{.2em}{-}x^{10}y^2$$
$$b)\hspace{.2em}x^{14}y^{14}$$
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#4:
Solutions:
$$a)\hspace{.2em}x^{21}y$$
$$b)\hspace{.2em}1$$
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#5:
Solutions:
$$a)\hspace{.2em}\frac{x^{10}y^8}{z^4}$$
$$b)\hspace{.2em}\frac{2^7x^{19}y^{25}}{z^6w^9}$$
