Question

use the rules of exponents to simplify the expression.
$(u^{6}v)(7v^{5})$
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use the rules of exponents to simplify the expression.
$(x^{9}y^{7})(6y^{7})$
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use the rules of exponents to simplify the expression.
$-8(y^{8})^{2}$
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Explanation:

Problem 1: Simplify \((u^{6}v)(7v^{5})\)

Step1: Multiply coefficients and like bases

Multiply the coefficient \(1\) (from \(u^6v\)) and \(7\), and use the product rule for exponents \(a^m \cdot a^n = a^{m + n}\) for \(v\) terms.
\((u^{6}v)(7v^{5}) = 7 \cdot u^{6} \cdot (v \cdot v^{5})\)

Step2: Apply exponent rule to \(v\)

For \(v \cdot v^{5}\), using \(a^m \cdot a^n = a^{m + n}\) with \(m = 1\) and \(n = 5\), we get \(v^{1+5}=v^{6}\).
So the expression becomes \(7u^{6}v^{6}\)

Step1: Multiply coefficients and like bases

Multiply the coefficient \(1\) (from \(x^9y^7\)) and \(6\), and use the product rule for exponents \(a^m \cdot a^n = a^{m + n}\) for \(y\) terms.
\((x^{9}y^{7})(6y^{7}) = 6 \cdot x^{9} \cdot (y^{7} \cdot y^{7})\)

Step2: Apply exponent rule to \(y\)

For \(y^{7} \cdot y^{7}\), using \(a^m \cdot a^n = a^{m + n}\) with \(m = 7\) and \(n = 7\), we get \(y^{7 + 7}=y^{14}\).
So the expression becomes \(6x^{9}y^{14}\)

Step1: Apply power of a power rule

Use the power of a power rule \((a^m)^n = a^{m \cdot n}\) for the \(y\) term.
\(-8(y^{8})^{2}=-8\cdot y^{8\times2}\)

Step2: Calculate the exponent

Calculate \(8\times2 = 16\), so the expression becomes \(-8y^{16}\)

Answer:

\(7u^{6}v^{6}\)

Problem 2: Simplify \((x^{9}y^{7})(6y^{7})\)