5.4 Radical functions (simplification)

1. Question

Find integers [math]a, b, c, d[/math], and [math]e[/math] such that [math]\sqrt[3]{54(xy^2)^5}=3^ax^by^c\sqrt[3]{2x^dy^e}[/math].

[math]a=[/math]
[math]b=[/math]
[math]c=[/math]
[math]d=[/math]
[math]e=[/math]

2. Question

Which of the following expressions is equivalent to [math]\sqrt{9x^4-6x^2+1}[/math]?

[math]3x^2-1[/math]
[math]-(3x^2-1)[/math]
[math]\pm(3x^2-1)[/math]
[math]|3x^2-1|[/math]
[math]-|3x^2-1|[/math]

3. Question

If [math]\displaystyle{\frac{x}{\sqrt[3]{x}-2}}=\dfrac{x^a+2x^b+4x^c}{x-8}[/math] where [math]a, b[/math], and [math]c[/math] are constants, what is the value of [math]a+b+c[/math]?

4. Question

If [math]\displaystyle{\frac{1}{1+\sqrt{2}+\sqrt{3}}}=\dfrac{a+\sqrt{b}-\sqrt{c}}{4}[/math] where [math]a, b[/math], and [math]c[/math] are positive integers, what is the value of [math]a+b+c[/math]?

5. Question

Which of the following expressions is equivalent to [math]\sqrt[4]{48x^4y^8z^9}[/math]?

[math]2|x|y^2z^2\sqrt[4]{3z}[/math]
[math]2xy^2z^2\sqrt[4]{3z}[/math]
[math]\pm2xy^2z^2\sqrt[4]{3z}[/math]
[math]\pm2|x|y^2z^2\sqrt[4]{3z}[/math]

6. Question

Find integers [math]x, y[/math], and [math]z[/math] such that [math]\sqrt{(5a)^3(3b)^5}=xa^yb^z\sqrt{15ab}[/math].

[math]x=[/math]
[math]y=[/math]
[math]z=[/math]

7. Question

If [math]\dfrac{8}{\sqrt{7}+\sqrt{3}}=a\sqrt{7}+b\sqrt{3}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a+b[/math]?

8. Question

If [math]\dfrac{2}{\sqrt{2}-1}+\dfrac{2}{\sqrt{3}+\sqrt{2}}=a+b\sqrt{3}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a+b[/math]?

9. Question

Which of the following expressions is equivalent to [math]\displaystyle{\sqrt{\frac{32x^3}{9y^{-5}}}}[/math]?

[math]\dfrac{4}{3}xy^2\sqrt{2xy}[/math]
[math]\pm\dfrac{4}{3}xy^2\sqrt{2xy}[/math]
[math]\pm\dfrac{4}{3}xy^{-2}\sqrt{2xy^{-1}}[/math]
[math]\dfrac{4}{3}xy^{-2}\sqrt{2xy^{-1}}[/math]
None of the above

10. Question

If [math]\displaystyle{(\sqrt{2022}-\sqrt{2021})^{2021}(\sqrt{2022}+\sqrt{2021})^{2022}}=\sqrt{a}+\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers and [math]a>b[/math], what is the value of [math]a[/math]?

11. Question

Find integers [math]p, q, r[/math], and [math]s[/math] such that [math]\displaystyle{\left(\sqrt[3]{ab^2c^4d^8}\right)^2}=bc^pd^q\cdot\sqrt[3]{a^2bc^rd^s}[/math].

[math]p=[/math]
[math]q=[/math]
[math]r=[/math]
[math]s=[/math]

12. Question

If [math]\displaystyle{\frac{\sqrt{3}-1}{\sqrt{3}+1}}=a-\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers, what is the value of [math]a+b[/math]?

13. Question

Which of the following expressions is equivalent to [math]\sqrt[4]{96(xy^2)^5}[/math]?

[math]2y^2\sqrt[4]{6xy^2}[/math]
[math]2xy\sqrt[4]{6xy^3}[/math]
[math]2xy^2\sqrt[4]{6xy^2}[/math]
[math]2y\sqrt[4]{6xy^3}[/math]
None of the above

14. Question

Which of the following expressions is equivalent to [math]-\sqrt{x^4-6x^2+9}[/math]?

[math]x^2-3[/math]
[math]-(x^2-3)[/math]
[math]|x^2-3|[/math]
[math]-|x^2-3|[/math]
None of the above

15. Question

If [math]\dfrac{a}{\sqrt{x+a}-\sqrt{x}}=k(\sqrt{x+a}+\sqrt{x})[/math] where [math]k[/math] is a constant, what is [math]k[/math]?

[math]1[/math]
[math]a[/math]
[math]-a[/math]
[math]a^2[/math]
None of the above

16. Question

The graph of which function is shown?

[math]y=4\sqrt{x}-3[/math]
[math]y=4\sqrt{x}+3[/math]
[math]y=\sqrt{4x}-3[/math]
[math]y=4\sqrt{x-3}[/math]
[math]y=\dfrac{1}{4}\sqrt{x}-3[/math]

17. Question

Express[math]\sqrt[5]{-3072s^{10}q^{15}}[/math] in simplified form.

[math]-4s^2q^3\sqrt[5]{3}[/math]
[math]-16s^2q^3\sqrt[5]{3}[/math]
[math]-6s^2q^3\sqrt[5]{8}[/math]
[math]-4s^3q^2\sqrt[5]{5}[/math]
None of the above

18. Question

Simplify [math]\dfrac{\sqrt{x}-3}{\sqrt{x}}[/math]

[math]\dfrac{x-3}{x}[/math]
[math]\dfrac{x-\sqrt{3}}{x}[/math]
[math]\dfrac{x-3\sqrt{x}}{x}[/math]
[math]\dfrac{x-\sqrt{3x}}{x}[/math]
None of the above

19. Question

Write [math](\sqrt[3]{8a^{-3}})^{-4}[/math] in simplest form.

[math]\displaystyle{\frac{a^4}{16}}[/math]
[math]\displaystyle{\frac{16}{a^4}}[/math]
[math]16a^4[/math]
[math]\displaystyle{\frac{1}{16a^4}}[/math]
None of the above.

20. Question

Evalaute [math]\sqrt[3]{(-27)^2}[/math].

21. Question

If [math]\sqrt{3x^2-18x+27}=ax+b[/math] where [math]a[/math] and [math]b[/math] are constants and [math]x<0[/math], what is the value of [math]a[/math]?

[math]\sqrt{3}[/math]
[math]-\sqrt{3}[/math]
[math]3\sqrt{3}[/math]
[math]-3\sqrt{3}[/math]
[math]\pm\sqrt{3}[/math]

22. Question

What is the simplified form of the expression [math]\displaystyle{\frac{8}{6+\sqrt{2w}}}[/math]?

[math]\displaystyle{\frac{4(1-\sqrt{2w})}{3-w}}[/math]
[math]\displaystyle{\frac{4(6-\sqrt{2w})}{3-w}}[/math]
[math]\displaystyle{\frac{8(6+\sqrt{2w})}{6+2w}}[/math]
[math]\displaystyle{\frac{4(6-\sqrt{2w})}{18-w}}[/math]

23. Question

The expression [math]\sqrt{50x^2y^9z^3}[/math] is equivalent to which of the following?

[math]5|x|y^3|z|\sqrt{2yz}[/math]
[math]5|x|y^4|z|\sqrt{2yz}[/math]
[math]2|x|y^3|z|\sqrt{5yz}[/math]
[math]2|x|y^4|z|\sqrt{5yz}[/math]

24. Question

What is the simplest form of the expression [math](\sqrt[5]{9\cdot3^3})^{-1}[/math]?

[math]-243[/math]
[math]-\displaystyle{\frac{1}{3}}[/math]
[math]\displaystyle{\frac{1}{3}}[/math]
[math]3[/math]

25. Question

What is the simplified form of the expression [math]-2\sqrt{200}-\sqrt{75}-3\sqrt{2}-5\sqrt{27}[/math]?

[math]-23\sqrt{2}-20\sqrt{3}[/math]
[math]-23\sqrt{2}-10\sqrt{3}[/math]
[math]-17\sqrt{2}-20\sqrt{3}[/math]
[math]-5\sqrt{2}-10\sqrt{3}[/math]

26. Question

What is the simplified form of the expression [math]\sqrt[5]{\displaystyle{\frac{-32m^{15}}{243n^{10}}}}[/math]?

[math]-\displaystyle{\frac{m^3}{8n^2}}[/math]
[math]-\displaystyle{\frac{2m\cdot\sqrt[5]{m}}{n^2}}[/math]
[math]-\displaystyle{\frac{m\cdot\sqrt[5]{m}}{8n^2}}[/math]
[math]-\displaystyle{\frac{2m^3}{3n^2}}[/math]

27. Question

What is the productof the expression [math]\sqrt{6}\cdot\sqrt[3]{9}[/math]?

[math]6\sqrt[3]{12}[/math]
[math]3\sqrt[6]{48}[/math]
[math]3\sqrt[6]{24}[/math]
[math]6\sqrt[3]{48}[/math]

28. Question

Simplify [math](4-\sqrt{7})^{-1}[/math]

[math]\displaystyle{\frac{4+\sqrt{7}}{-3}}[/math]
[math]\displaystyle{\frac{4+\sqrt{7}}{9}}[/math]
[math]\displaystyle{\frac{4-\sqrt{7}}{3}}[/math]
[math]\displaystyle{\frac{4-\sqrt{7}}{9}}[/math]

29. Question

What is [math](\sqrt[3]{6h}+2)(\sqrt[3]{6h}-7)[/math]?

[math]\sqrt[3]{36h^2}-5\sqrt[3]{6h}-14[/math]
[math]6\sqrt[3]{h^2}+5\sqrt[3]{6h}-14[/math]
[math]6\sqrt[3]{h^2}-5\sqrt[3]{6h}-14[/math]
[math]2\sqrt[3]{h^2}-5\sqrt[3]{6h}-14[/math]

30. Question

Simplify the product completely: [math]\sqrt{6x^5y^2}\cdot\sqrt{3xy^3}[/math]

[math]2x^2y^3\sqrt{3y}[/math]
[math]3x^3y^2\sqrt{2y}[/math]
[math]3x^2y\sqrt{2y}[/math]
[math]2x^3y^2\sqrt{3y}[/math]