Exponents and Radicals Exercises

Subsection 2.5 Exponents and Radicals Exercises

Let $p(x)=x^2\text{,}$ $q(x)=-5^x\text{,}$ $r(x)=5^{x-3}\text{.}$ Write a formula for each function below, using the exponent rules to simplify the formula and write without negative exponents.
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1. $\displaystyle a(x)=p(q(x))$
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2. $\displaystyle b(x)=2q(x)r(x)$
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3. $\displaystyle c(x)=(q(x))^{-3}$
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4. $\displaystyle d(x)=5\left(\frac{q(x)}{r(x)}\right)^3$
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Let $f(x)=\sqrt{x^2}\text{,}$ $g(x)=\sqrt[3]{x^3}\text{,}$ $h(x)=\sqrt[4]{x}\text{,}$ and $k(x)=x^4\text{.}$
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1. True or false: For all real values $x\text{,}$ $f(x)=g(x)\text{.}$ Explain your answer.
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2. True or false: For all real values $x \geq 0\text{,}$ $f(x)=k(h(x))\text{.}$ Explain your answer.
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3. True or false: For all real values $x \geq 0\text{,}$ $h(k(x))=k(h(x))\text{.}$ Explain your answer.
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4. True or false: For all real values $x\text{,}$ $f(x)=h(k(x))\text{.}$ Explain your answer.
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Let $m(x)=x^{\frac{1}{6}}\text{,}$ $n(x)=x^3\text{,}$ and $p(x)=x^4\text{.}$
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1. Write $m(n(x))$ as a function of $x$ with a single exponent.
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2. Write $m(n(x)p(x))$ as a function of $x$ with a single exponent.
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3. For what values of $x$ is $m(p(x))=x^{\frac{2}{3}}\text{?}$ Explain.
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4. For what values of $x$ is $\sqrt[3]{n(x)}=x\text{?}$ Explain.
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1. For what values of $x$ is $3\cdot2^x=6^x\text{?}$ Explain.
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2. Is it ok to simplify $3\cdot2^x$ into $6^x\text{?}$ Explain.
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Write each expression as a sum of powers.
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1. $\displaystyle \frac{x^3-4x^2+7x}{\sqrt{x}}$
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2. $\displaystyle x^{\frac{2}{3}}(3x^5-6x^{\frac{5}{3}}-9x\sqrt[3]{x})$
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3. $\displaystyle x^{\frac{5}{2}}(-8x^2+5x^{\frac{3}{2}}-10\sqrt{x}+4)$
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Let $g(x)=-\frac{1}{2}x^2+4x-6\text{.}$
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1. Find the $x-$ and $y-$intercepts of $g(x)\text{.}$
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2. Find the vertex of $g(x)\text{.}$
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3. Use Item 6.a and Item 6.b to graph $g(x)\text{.}$ Use Desmos to check your answer.
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4. Solve algebraically: $g(x)=-2.5\text{.}$ Use Desmos to check your answer.
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5. Solve algebraically: $g(x)=1.5\text{.}$ Use Desmos to check your answer.
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6. Solve $g(x) < -6\text{.}$
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The graph of $b(n)$ is shown below.
$A coordinate grid with horizontal axis labeled n and vertical axis labeled b(n). The graph shows an upward-opening parabola. The curve has a minimum at $(2,-2)\text{,}$ then increases on both sides.$

Figure 2.5.7. Graph of the function $b(n)\text{.}$
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1. Evaluate $b(0)\text{.}$
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2. Solve $b(n)=5\text{.}$
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3. Solve $b(n) \leq 5\text{.}$
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4. Write a function equation for $b(n)\text{.}$
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5. Use your equation to find the $n-$intercepts of $b\text{.}$
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The number of computer science degrees awarded by Monroe College has increased by a factor of 1.5 every 5 years since 1984. The college granted 8 degrees in 1984.
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1. Write a formula for the number of degrees awarded $t$ years after 1984.
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2. If the number of degrees awarded continued to increase exponentially, how many degrees were awarded in 2000?
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3. If the number of degrees awarded continues to increase exponentially, in what year will 200 degrees be awarded?
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