Sure! Let's simplify the expression step by step:
[tex]\[
\frac{\left(2^{10}\right)^3 \cdot 2^{-10}}{2^{-7}}
\][/tex]
### Step 1: Simplify [tex]\( (2^{10})^3 \)[/tex]
First, simplify the part inside the parentheses:
[tex]\[
(2^{10})^3
\][/tex]
Using the power of a power property of exponents, we multiply the exponents:
[tex]\[
2^{10 \cdot 3} = 2^{30}
\][/tex]
### Step 2: Combine [tex]\( 2^{30} \cdot 2^{-10} \)[/tex]
Next, we need to multiply [tex]\( 2^{30} \)[/tex] by [tex]\( 2^{-10} \)[/tex]. Using the product of powers property of exponents (which says [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]):
[tex]\[
2^{30} \cdot 2^{-10} = 2^{30 - 10} = 2^{20}
\][/tex]
### Step 3: Divide by [tex]\( 2^{-7} \)[/tex]
Now, we need to divide [tex]\( 2^{20} \)[/tex] by [tex]\( 2^{-7} \)[/tex]. Using the quotient of powers property of exponents (which says [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]):
[tex]\[
\frac{2^{20}}{2^{-7}} = 2^{20 - (-7)} = 2^{20 + 7} = 2^{27}
\][/tex]
So, the simplified expression is:
[tex]\[
2^{27}
\][/tex]
### Final Value Calculation
The value of [tex]\( 2^{27} \)[/tex] is:
[tex]\[
2^{27} = 134217728
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{\left(2^{10}\right)^3 \cdot 2^{-10}}{2^{-7}} = 2^{27} = 134217728
\][/tex]
