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Chapter 2: Problem 18

Simplify the given expression by writing it as a power of a single variable. $$ x\left(x^{4}\left(x^{3}\right)^{2}\right)^{5 / 3} $$

Short Answer

Expert verified
The simplified expression as a power of a single variable is \(x^{\frac{53}{3}}\).

Step by step solution

01

Apply the power of a power rule.

We need to apply the power of a power rule to simplify the expression: \((a^m)^n = a^{m*n}\) In our exercise, we have \((x^3)^2\), thus apply the rule to this term.
02

Simplify the expression inside the bracket.

Now that you have applied the power of a power rule, the expression will look like this: \[ x\left(x^4 \cdot x^6\right)^{5/3} \] Now, we have \(x^4 \cdot x^6\) inside the bracket that can be simplified using product rule of exponents.
03

Apply the product rule of exponents.

Product rule of exponents states that: \(a^m \cdot a^n = a^{m+n}\) Apply this rule to the expression inside the bracket which is \(x^4 \cdot x^6\)
04

Rewrite the expression with the simplified term.

After applying the product rule of exponents, rewrite the expression accordingly: \[ x \left(x^{10}\right)^{5/3} \]
05

Apply the power of a power rule again.

Now, we need to apply the power of a power rule to the remaining exponent term. In our case, it is \(\left(x^{10}\right)^{5/3}\). Apply the rule to this term.
06

Apply the product rule of exponents again.

Now we have the expression: \[ x \cdot x^{\frac{50}{3}} \] Apply the product rule of exponents to \(x \cdot x^{\frac{50}{3}}\).
07

Write the final simplified expression.

After applying both the power of a power rule and the product rule of exponents, the final simplified expression is: \[ x^{\frac{53}{3}} \] This is the simplified expression as a power of a single variable.

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Most popular questions from this chapter

Suppose $$r(x)=\frac{x+1}{x^{2}+3} \quad \text { and } \quad s(x)=\frac{x+2}{x^{2}+5}$$ Find two distinct numbers \(x\) such that \(s(x)=\frac{1}{8}\).

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

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

Without doing any calculations or using a calculator, explain why $$ x^{2}+87559743 x-787727821 $$ has no integer zeros. [Hint: If \(x\) is an odd integer, is the expression above even or odd? If \(x\) is an even integer, is the expression above even or odd?]

Show that if \(p\) and \(q\) are nonzero polynomials, then $$ \operatorname{deg}(p \circ q)=(\operatorname{deg} p)(\operatorname{deg} q) $$.
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