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Answer :
To simplify the expression [tex]\(2x y^8 \cdot 7 w^{-8} w^8 y^{-9} \cdot 4x^7\)[/tex] and use only positive exponents in your answer, we'll follow these steps:
1. Multiply the coefficients: The coefficients in the expression are 2, 7, and 4. Multiply these together:
[tex]\[
2 \times 7 \times 4 = 56
\][/tex]
2. Combine the exponents for [tex]\(x\)[/tex]: The expression includes [tex]\(x\)[/tex] (which is the same as [tex]\(x^1\)[/tex]) and [tex]\(x^7\)[/tex]. When you multiply terms with the same base, you add the exponents:
[tex]\[
x^1 \times x^7 = x^{1+7} = x^8
\][/tex]
3. Combine the exponents for [tex]\(y\)[/tex]: The expression includes [tex]\(y^8\)[/tex] and [tex]\(y^{-9}\)[/tex]. Again, add the exponents:
[tex]\[
y^8 \times y^{-9} = y^{8-9} = y^{-1}
\][/tex]
4. Combine the exponents for [tex]\(w\)[/tex]: The expression contains [tex]\(w^{-8}\)[/tex] and [tex]\(w^8\)[/tex]. Add these exponents:
[tex]\[
w^{-8} \times w^8 = w^{-8+8} = w^0
\][/tex]
Since any base raised to the power of 0 is 1, [tex]\(w^0\)[/tex] can be omitted from the product.
5. Write the simplified expression: Substitute back into the expression:
[tex]\[
56 \cdot x^8 \cdot y^{-1}
\][/tex]
6. Use only positive exponents: Convert [tex]\(y^{-1}\)[/tex] into a positive exponent by moving it to the denominator:
[tex]\[
\frac{56 \cdot x^8}{y}
\][/tex]
The final simplified expression, with only positive exponents, is:
[tex]\[
\frac{56x^8}{y}
\][/tex]
1. Multiply the coefficients: The coefficients in the expression are 2, 7, and 4. Multiply these together:
[tex]\[
2 \times 7 \times 4 = 56
\][/tex]
2. Combine the exponents for [tex]\(x\)[/tex]: The expression includes [tex]\(x\)[/tex] (which is the same as [tex]\(x^1\)[/tex]) and [tex]\(x^7\)[/tex]. When you multiply terms with the same base, you add the exponents:
[tex]\[
x^1 \times x^7 = x^{1+7} = x^8
\][/tex]
3. Combine the exponents for [tex]\(y\)[/tex]: The expression includes [tex]\(y^8\)[/tex] and [tex]\(y^{-9}\)[/tex]. Again, add the exponents:
[tex]\[
y^8 \times y^{-9} = y^{8-9} = y^{-1}
\][/tex]
4. Combine the exponents for [tex]\(w\)[/tex]: The expression contains [tex]\(w^{-8}\)[/tex] and [tex]\(w^8\)[/tex]. Add these exponents:
[tex]\[
w^{-8} \times w^8 = w^{-8+8} = w^0
\][/tex]
Since any base raised to the power of 0 is 1, [tex]\(w^0\)[/tex] can be omitted from the product.
5. Write the simplified expression: Substitute back into the expression:
[tex]\[
56 \cdot x^8 \cdot y^{-1}
\][/tex]
6. Use only positive exponents: Convert [tex]\(y^{-1}\)[/tex] into a positive exponent by moving it to the denominator:
[tex]\[
\frac{56 \cdot x^8}{y}
\][/tex]
The final simplified expression, with only positive exponents, is:
[tex]\[
\frac{56x^8}{y}
\][/tex]
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