Question

simplify the rational expression. \\(\frac{y^3 - 16y}{y^2 + 4y - 32}\\) (no response), \\(y \
eq 4\\) 3. - / 1 points multiply and simplify. \\(16u^4 \cdot \frac{12}{8u^2}\\) (no response), \\(u \
eq 0\\)

Explanation:

First Problem: Simplify \(\boldsymbol{\frac{y^3 - 16y}{y^2 + 4y - 32}}\)

Step1: Factor numerator and denominator

Factor \(y\) from the numerator: \(y^3 - 16y = y(y^2 - 16)\). Then, \(y^2 - 16\) is a difference of squares, so \(y^2 - 16=(y - 4)(y + 4)\). Thus, the numerator becomes \(y(y - 4)(y + 4)\).

Factor the denominator \(y^2 + 4y - 32\). We need two numbers that multiply to \(-32\) and add to \(4\). Those numbers are \(8\) and \(-4\). So, \(y^2 + 4y - 32=(y + 8)(y - 4)\).

Now the expression is \(\frac{y(y - 4)(y + 4)}{(y + 8)(y - 4)}\).

Step2: Cancel common factors

Cancel the common factor \((y - 4)\) (since \(y
eq4\), we can do this). The simplified expression is \(\frac{y(y + 4)}{y + 8}\) or \(\frac{y^2 + 4y}{y + 8}\).

Step1: Multiply the coefficients and use exponent rules

First, multiply the coefficients: \(16\times\frac{12}{8}\). \(16\div8 = 2\), then \(2\times12 = 24\).

For the variables, use the rule \(a^m\cdot a^n=a^{m + n}\) (but here we have division in terms of exponents: \(u^4\div u^2=u^{4 - 2}\) since \(\frac{u^4}{u^2}=u^{4-2}\)).

So, \(16u^4\cdot\frac{12}{8u^2}=\frac{16\times12}{8}u^{4 - 2}\).

Step2: Simplify the result

Simplify the coefficient: \(\frac{16\times12}{8}=24\). Simplify the exponent: \(u^{4 - 2}=u^2\). So the simplified expression is \(24u^2\).

Answer:

\(\frac{y^2 + 4y}{y + 8}\) (or \(\frac{y(y + 4)}{y + 8}\))

Second Problem: Simplify \(\boldsymbol{16u^4\cdot\frac{12}{8u^2}}\)