Rewriting Expressions with Radicals, Rational Exponents - Algebra 1

1. What is $\sqrt[3]{a}\cdot \sqrt[6]{a}$ rewritten as a single rational exponent?

   $a^{\frac{1}{2}}$. Add exponents: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$.

2. What is $27^{\frac{1}{3}}$ rewritten as a radical and simplified?

   $3$. $\sqrt[3]{27} = 3$ since $3^3 = 27$.

3. What is the power of a quotient rule written in exponent form?

   $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$. Distribute the exponent to numerator and denominator.

4. What is the power of a product rule written in exponent form?

   $(ab)^n=a^n b^n$. Distribute the exponent to each factor in the product.

5. What is $\frac{\sqrt[3]{x^4}}{\sqrt[3]{x}}$ simplified using rational exponents?

   $x$. Subtract exponents: $\frac{4}{3} - \frac{1}{3} = 1$, so $x^1 = x$.

6. What is $\left(\sqrt[3]{x^2}\right)^{\frac{3}{2}}$ simplified using exponent rules?

   $x$. Apply power rule: $(x^{\frac{2}{3}})^{\frac{3}{2}} = x^1 = x$.

7. What is $64^{\frac{5}{6}}$ simplified?

   $32$. $64^{\frac{5}{6}} = (2^6)^{\frac{5}{6}} = 2^5 = 32$.

8. What is $\left(x^{\frac{2}{5}}\cdot x^{\frac{1}{5}}\right)^5$ simplified?

   $x^3$. Add exponents inside, then apply outer power: $(x^{\frac{3}{5}})^5 = x^3$.

9. What is $\left(x^{-\frac{1}{2}}\right)^4$ simplified?

   $\frac{1}{x^2}$. Multiply exponents: $-\frac{1}{2} \cdot 4 = -2$, so $x^{-2} = \frac{1}{x^2}$

10. What is $(x^{\frac{2}{3}})^3$ simplified?

    $x^2$. Multiply exponents: $\frac{2}{3} \cdot 3 = 2$.

11. What is $\left(\frac{a^{\frac{1}{2}}}{a^{\frac{1}{4}}}\right)$ simplified?

    $a^{\frac{1}{4}}$. Subtract exponents: $\frac{1}{2} - \frac{1}{4} = \frac{1}{4}$.

12. What is $\sqrt[3]{x^2}\div \sqrt[3]{x^5}$ simplified using exponents?

    $\frac{1}{x}$. Subtract exponents: $\frac{2}{3} - \frac{5}{3} = -1$, so $x^{-1} = \frac{1}{x}$.

13. What is $\frac{1}{x^{\frac{5}{2}}}$ rewritten using a negative exponent?

    $x^{-\frac{5}{2}}$. Reciprocal becomes negative exponent in numerator.

14. What is $\sqrt[4]{x^3}\cdot \sqrt[4]{x}$ simplified using exponents?

    $x$. Add exponents: $\frac{3}{4} + \frac{1}{4} = 1$, so $x^1 = x$.

15. What is $\sqrt[3]{a^2}\cdot \sqrt[3]{a}$ simplified using exponents?

    $a$. Add exponents: $\frac{2}{3} + \frac{1}{3} = 1$, so $a^1 = a$.

16. What is $81^{\frac{3}{4}}$ simplified?

    $27$. $81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27$.

17. What is the product rule for exponents written in exponent form for the same base $a$?

    $a^m\cdot a^n=a^{m+n}$. When multiplying same bases, add the exponents.

18. What is $32^{\frac{2}{5}}$ simplified?

    $4$. $32^{\frac{2}{5}} = (2^5)^{\frac{2}{5}} = 2^2 = 4$.

19. What is $16^{\frac{1}{2}}$ rewritten as a radical and simplified?

    $4$. $\sqrt{16} = 4$ since $4^2 = 16$.

20. What is $(x^{\frac{3}{4}})^2$ simplified?

    $x^{\frac{3}{2}}$. Multiply exponents: $\frac{3}{4} \cdot 2 = \frac{3}{2}$.

21. What is $\sqrt[6]{x^3}$ rewritten using a rational exponent and simplified?

    $x^{\frac{1}{2}}$. Simplify the fraction: $\frac{3}{6} = \frac{1}{2}$.

22. What is $\left(a^{\frac{3}{2}}\right)\left(a^{-\frac{1}{2}}\right)$ simplified?

    $a$. Add exponents: $\frac{3}{2} + (-\frac{1}{2}) = 1$.

23. What is $\sqrt{x}\cdot x^{\frac{3}{2}}$ simplified using exponent rules?

    $x^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$.

24. What is $x^{-\frac{2}{3}}$ rewritten without negative exponents?

    $\frac{1}{x^{\frac{2}{3}}}$. Negative exponent moves to denominator as positive.

25. What is $\sqrt[n]{a}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?

    $a^{\frac{1}{n}}$. By definition, $\sqrt[n]{a} = a^{\frac{1}{n}}$.

26. What is $\sqrt[8]{x^4}$ rewritten using a simplified rational exponent?

    $x^{\frac{1}{2}}$. Simplify the fraction: $\frac{4}{8} = \frac{1}{2}$.

27. What is $\sqrt{x^5}$ rewritten using rational exponents?

    $x^{\frac{5}{2}}$. Convert radical to rational exponent form.

28. What is $\left(\sqrt[5]{a}\right)^3$ rewritten using a rational exponent?

    $a^{\frac{3}{5}}$. Convert radical to rational exponent and apply power.

29. What is $\left(\sqrt{x}\right)^3$ rewritten using a rational exponent and simplified?

    $x^{\frac{3}{2}}$. Apply power rule: $(x^{\frac{1}{2}})^3 = x^{\frac{3}{2}}$.

30. What is $\frac{x^{\frac{5}{3}}}{x^{\frac{2}{3}}}$ simplified?

    $x$. Subtract exponents: $\frac{5}{3} - \frac{2}{3} = 1$.

31. What is $\sqrt{a}\cdot \sqrt{a^3}$ simplified using rational exponents?

    $a^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$, so $a^2$.

32. What is $\sqrt[3]{x^2}$ rewritten using a rational exponent?

    $x^{\frac{2}{3}}$. Convert radical to rational exponent form.

33. What is $\left(\sqrt[3]{x}\right)^{-2}$ rewritten without negative exponents?

    $\frac{1}{x^{\frac{2}{3}}}$. Negative exponent moves to denominator as positive.

34. What is $\sqrt[n]{a^m}$ written using rational exponents (assume $a\ge 0$ and $n\in\mathbb{N}$)?

    $a^{\frac{m}{n}}$. By definition, $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.

35. What is $a^{\frac{m}{n}}$ written as a radical (assume $a\ge 0$, $m\in\mathbb{Z}$, $n\in\mathbb{N}$)?

    $\sqrt[n]{a^m}$. By definition, $a^{\frac{m}{n}} = \sqrt[n]{a^m}$.

36. What is the quotient rule for exponents written in exponent form for the same base $a$?

    $\frac{a^m}{a^n}=a^{m-n}$. When dividing same bases, subtract the exponents.

37. What is the power of a power rule for exponents written in exponent form?

    $(a^m)^n=a^{mn}$. When raising a power to a power, multiply exponents.

38. What is the negative exponent rule written in exponent form (assume $a\ne 0$)?

    $a^{-n}=\frac{1}{a^n}$. Negative exponents become reciprocals with positive exponents.

39. What is $\left(x^3\right)^{\frac{1}{2}}$ rewritten using a single rational exponent?

    $x^{\frac{3}{2}}$. Power of a power rule: multiply exponents.

40. What is $a^{0}$ for $a\ne 0$?

    $1$. Any nonzero base to the zero power equals 1.

41. What is $\frac{\sqrt[4]{x}}{\sqrt{x}}$ rewritten using rational exponents and simplified?

    $\frac{1}{x^{\frac{1}{4}}}$. Subtract exponents: $\frac{1}{4} - \frac{1}{2} = -\frac{1}{4}$.

42. What is $x^{\frac{1}{2}}\cdot x^{\frac{3}{2}}$ simplified?

    $x^2$. Add exponents: $\frac{1}{2} + \frac{3}{2} = 2$.

43. What is $x^{\frac{7}{3}}\div x^{\frac{1}{3}}$ simplified?

    $x^2$. Subtract exponents: $\frac{7}{3} - \frac{1}{3} = 2$.

44. What is $\left(\frac{1}{a^{\frac{2}{3}}}\right)\left(a^{\frac{5}{3}}\right)$ simplified?

    $a$. Multiply by reciprocal: add exponents $-\frac{2}{3} + \frac{5}{3} = 1$.

45. What is $\sqrt[4]{x^7}$ rewritten using rational exponents?

    $x^{\frac{7}{4}}$. Convert radical to rational exponent form.

46. What is $x^{\frac{4}{6}}$ simplified to an equivalent rational exponent in lowest terms?

    $x^{\frac{2}{3}}$. Simplify the fraction: $\frac{4}{6} = \frac{2}{3}$.

47. What is $\left(\frac{x^{\frac{1}{2}}}{x^{\frac{3}{2}}}\right)$ simplified?

    $\frac{1}{x}$. Subtract exponents: $\frac{1}{2} - \frac{3}{2} = -1$, so $\frac{1}{x}$.

48. What is $a^{\frac{1}{n}}$ written as a radical (assume $a\ge 0$ and $n\in\mathbb{N}$)?

    $\sqrt[n]{a}$. By definition, $a^{\frac{1}{n}} = \sqrt[n]{a}$.

49. What is $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{2}{3}}\right)$ simplified?

    $x$. Add exponents: $\frac{1}{3} + \frac{2}{3} = 1$.

50. What is $\left(\frac{x}{y}\right)^{\frac{3}{2}}$ rewritten as a radical?

    $\sqrt{\left(\frac{x}{y}\right)^3}$. Convert rational exponent to radical form.

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