Answer :
Sure, let's simplify the expression step by step:
The expression given is:
[tex]\[ 7v^{-7} \cdot 3x^6u^7u^{-7}v \cdot 2x^8 \][/tex]
We'll break it down:
1. Simplify [tex]\( u^7 \cdot u^{-7} \)[/tex]:
When you multiply powers with the same base, you add the exponents:
[tex]\[
u^7 \cdot u^{-7} = u^{7 + (-7)} = u^0 = 1
\][/tex]
2. Simplify [tex]\( v^{-7} \cdot v \)[/tex]:
Add the exponents:
[tex]\[
v^{-7} \cdot v = v^{-7 + 1} = v^{-6}
\][/tex]
3. Simplify [tex]\( x^6 \cdot x^8 \)[/tex]:
Again, add the exponents:
[tex]\[
x^6 \cdot x^8 = x^{6 + 8} = x^{14}
\][/tex]
4. Combine the constants:
Multiply the numerical coefficients:
[tex]\[
7 \cdot 3 \cdot 2 = 42
\][/tex]
5. Put it all together:
Combine all the simplified parts:
[tex]\[
42 \cdot x^{14} \cdot 1 \cdot v^{-6} = 42x^{14}v^{-6}
\][/tex]
6. Rewrite with positive exponents:
To express with positive exponents, rewrite [tex]\( v^{-6} \)[/tex]:
[tex]\[
v^{-6} = \frac{1}{v^6}
\][/tex]
So the entire expression becomes:
[tex]\[
42x^{14} \cdot \frac{1}{v^6} = \frac{42x^{14}}{v^6}
\][/tex]
Thus, the simplified expression using only positive exponents is:
[tex]\[ \frac{42x^{14}}{v^6} \][/tex]
The expression given is:
[tex]\[ 7v^{-7} \cdot 3x^6u^7u^{-7}v \cdot 2x^8 \][/tex]
We'll break it down:
1. Simplify [tex]\( u^7 \cdot u^{-7} \)[/tex]:
When you multiply powers with the same base, you add the exponents:
[tex]\[
u^7 \cdot u^{-7} = u^{7 + (-7)} = u^0 = 1
\][/tex]
2. Simplify [tex]\( v^{-7} \cdot v \)[/tex]:
Add the exponents:
[tex]\[
v^{-7} \cdot v = v^{-7 + 1} = v^{-6}
\][/tex]
3. Simplify [tex]\( x^6 \cdot x^8 \)[/tex]:
Again, add the exponents:
[tex]\[
x^6 \cdot x^8 = x^{6 + 8} = x^{14}
\][/tex]
4. Combine the constants:
Multiply the numerical coefficients:
[tex]\[
7 \cdot 3 \cdot 2 = 42
\][/tex]
5. Put it all together:
Combine all the simplified parts:
[tex]\[
42 \cdot x^{14} \cdot 1 \cdot v^{-6} = 42x^{14}v^{-6}
\][/tex]
6. Rewrite with positive exponents:
To express with positive exponents, rewrite [tex]\( v^{-6} \)[/tex]:
[tex]\[
v^{-6} = \frac{1}{v^6}
\][/tex]
So the entire expression becomes:
[tex]\[
42x^{14} \cdot \frac{1}{v^6} = \frac{42x^{14}}{v^6}
\][/tex]
Thus, the simplified expression using only positive exponents is:
[tex]\[ \frac{42x^{14}}{v^6} \][/tex]
