/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 128 Show that \(3^{3 / 2} 12^{3 / 2}... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 128

Show that \(3^{3 / 2} 12^{3 / 2}=216\)

Short Answer

Expert verified
Using the product rule for exponents, we rewrite the given expression as \((3 \cdot 12)^{\frac{3}{2}}\). Simplifying the product inside the parentheses, we get \(36^{\frac{3}{2}}\). Converting the base to 6, we get \((6^2)^{\frac{3}{2}}\). Applying the power rule for exponents, we get \(6^3\). Finally, calculating the expression, we find that \(3^{\frac{3}{2}} 12^{\frac{3}{2}} = 216\).

Step by step solution

01

Apply the product rule for exponents

To simplify the expression, we can apply the product rule for exponents which states that \(a^m b^m = (ab)^m\). In this case, we have \(3^{\frac{3}{2}} 12^{\frac{3}{2}}\), so \(m = \frac{3}{2}\) and \(a = 3\), \(b = 12\). Therefore, the expression can be rewritten as: \((3 \cdot 12)^{\frac{3}{2}}\)
02

Simplify the product inside the parentheses

Now, simplify the product inside the parentheses: \(3 \cdot 12 = 36\) Now the expression is: \(36^{\frac{3}{2}}\)
03

Apply the power rule for exponents

The power rule for exponents states that \((a^m)^n = a^{mn}\). In our case, we have \(36^{\frac{3}{2}}\), so \(a = 36\) and \(m = \frac{3}{2}\). We know that \(36 = 6^2\), so the expression can be rewritten as: \((6^2)^{\frac{3}{2}}\) Now let's apply the power rule: \(6^{(2 \cdot \frac{3}{2})}\)
04

Simplify the exponent multiplication

Simplify the exponent multiplication: \(6^{(2 \cdot \frac{3}{2})} = 6^3\)
05

Calculate the expression

Now, calculate the expression: \(6^3 = 6 \cdot 6 \cdot 6 = 216\) We have shown that \(3^{\frac{3}{2}} 12^{\frac{3}{2}} = 216\).

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Most popular questions from this chapter

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} $$

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{6 x^{4}+4 x^{3}-7}{2 x^{4}+3 x^{2}+5} $$

Suppose \(p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n},\) where \(a_{0}, a_{1}, \ldots, a_{n}\) are integers. Suppose \(M\) and \(N\) are nonzero integers with no common factors and \(\frac{M}{N}\) is a zero of \(p\). Show that \(a_{0} / M\) and \(a_{n} / N\) are integers. [Thus to find rational zeros of a polynomial with integer coefficients, we need only look at fractions whose numerator is a divisor of the constant term and whose denominator is a divisor of the coefficient of highest degree. This result is called the Rational Zeros Theorem or the Rational Roots Theorem.]

Find the asymptotes of the graph of the given function \(\mathrm{r}\). $$ r(x)=\frac{6 x^{6}-7 x^{3}+3}{3 x^{6}+5 x^{4}+x^{2}+1} $$

Suppose \(p\) and \(q\) are polynomials and the horizonal axis is an asymptote of the graph of \(\frac{p}{q}\). Explain why $$ \operatorname{deg} p<\operatorname{deg} q $$
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