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Math  /  Algebra

QuestionSelect all the expressions that are equivalent to 3−8⋅3−33^{-8} \cdot 3^{-3}. 3−113^{-11} 3243^{24} 3−833\frac{3^{-8}}{3^{3}} 13−11\frac{1}{3^{-11}}

Studdy Solution

STEP 1

1. The expressions involve properties of exponents.
2. We will use the property that am⋅an=am+n a^m \cdot a^n = a^{m+n} .
3. We will also use the property that aman=am−n \frac{a^m}{a^n} = a^{m-n} .
4. Negative exponents can be rewritten as reciprocals: a−m=1am a^{-m} = \frac{1}{a^m} .

STEP 2

1. Simplify the expression 3−8⋅3−3 3^{-8} \cdot 3^{-3} .
2. Compare the simplified expression to each given option.
3. Determine if each option is equivalent to the simplified expression.

STEP 3

Simplify the expression 3−8⋅3−3 3^{-8} \cdot 3^{-3} using the property of exponents:
3−8⋅3−3=3−8+(−3)=3−11 3^{-8} \cdot 3^{-3} = 3^{-8 + (-3)} = 3^{-11}

STEP 4

Compare the simplified expression 3−11 3^{-11} to each given option:
1. 3−11 3^{-11} : This is exactly the same as our simplified expression.
2. 324 3^{24} : This is not equivalent because the exponents are different.
3. 3−833 \frac{3^{-8}}{3^{3}} : Simplify using the property aman=am−n \frac{a^m}{a^n} = a^{m-n} :
3−833=3−8−3=3−11 \frac{3^{-8}}{3^{3}} = 3^{-8 - 3} = 3^{-11}
This is equivalent to 3−11 3^{-11} .
4. 13−11 \frac{1}{3^{-11}} : Simplify using the property of negative exponents, a−m=1am a^{-m} = \frac{1}{a^m} :
13−11=311 \frac{1}{3^{-11}} = 3^{11}
This is not equivalent to 3−11 3^{-11} .
The expressions that are equivalent to 3−8⋅3−3 3^{-8} \cdot 3^{-3} are 3−11 \boxed{3^{-11}} and 3−833 \boxed{\frac{3^{-8}}{3^{3}}} .

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