Question

name: period: algebra 2 final exam study guide (s1) your final exam for algebra 2 (semester 1) covers topics from unit 4. here are the topics cover - properties of exponents and radicals/ square and cube roots - the discriminant - solving rational equations - the number i - solving quadratics (completing the square) topic 1: properties of exponents and radicals/ square and cube roots write each expression in radical form 1) $7^{\frac{1}{2}}$ $sqrt{7^{-2}}$ 2) $4^{\frac{4}{3}}$ $sqrt4{4^{-3}}$ 3) $2^{\frac{5}{3}}$ $sqrt5{2^{-3}}$ 4) $7^{\frac{4}{3}}$ $sqrt4{7^{-3}}$ 5) $6^{\frac{3}{2}}$ $sqrt3{6^{-2}}$ 6) $2^{\frac{1}{6}}$ $sqrt{2^{-6}}$ write each expression in exponential form 7) $(sqrt{10})^{3}$ $10^{\frac{3}{x}}$ 8) $sqrt6{2}$ $2^{\frac{x}{6}}$ 9) $(sqrt4{2})^{5}$ $2^{\frac{5}{4}}$ 10) $(sqrt4{5})^{5}$ $5^{\frac{5}{4}}$ 11) $sqrt3{2}$ $2^{\frac{x}{3}}$ 12) $sqrt6{10}$ $10^{\frac{x}{6}}$

Explanation:

Step1: Recall the relationship between exponents and radicals

The formula to convert between exponential form \( a^{\frac{m}{n}} \) and radical form \( \sqrt[n]{a^m} \) (or \( (\sqrt[n]{a})^m \)) is used. For a number \( a \), \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \) and \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \).

Step2: Solve problem 1: \( 7^{\frac{1}{2}} \) to radical form

Using the formula \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), here \( a = 7 \), \( m = 1 \), \( n = 2 \). So \( 7^{\frac{1}{2}}=\sqrt[2]{7^1}=\sqrt{7} \). The given \( \sqrt{7^{-2}} \) might be a typo, but focusing on \( 7^{\frac{1}{2}} \), its radical form is \( \sqrt{7} \).

Step3: Solve problem 2: \( 4^{\frac{4}{3}} \) to radical form

Using \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), \( a = 4 \), \( m = 4 \), \( n = 3 \). So \( 4^{\frac{4}{3}}=\sqrt[3]{4^4} \).

Step4: Solve problem 3: \( 2^{\frac{5}{3}} \) to radical form

\( a = 2 \), \( m = 5 \), \( n = 3 \), so \( 2^{\frac{5}{3}}=\sqrt[3]{2^5} \).

Step5: Solve problem 4: \( 7^{\frac{4}{3}} \) to radical form

\( a = 7 \), \( m = 4 \), \( n = 3 \), so \( 7^{\frac{4}{3}}=\sqrt[3]{7^4} \).

Step6: Solve problem 5: \( 6^{\frac{3}{2}} \) to radical form

\( a = 6 \), \( m = 3 \), \( n = 2 \), so \( 6^{\frac{3}{2}}=\sqrt[2]{6^3}=\sqrt{6^3} \) or \( (\sqrt{6})^3 \).

Step7: Solve problem 6: \( 2^{\frac{1}{6}} \) to radical form

\( a = 2 \), \( m = 1 \), \( n = 6 \), so \( 2^{\frac{1}{6}}=\sqrt[6]{2^1}=\sqrt[6]{2} \). The \( \sqrt{2^6} \) might be a typo, but \( 2^{\frac{1}{6}} \) is \( \sqrt[6]{2} \).

Step8: Solve problem 7: \( (\sqrt{10})^3 \) to exponential form

\( \sqrt{10}=10^{\frac{1}{2}} \), so \( (\sqrt{10})^3=(10^{\frac{1}{2}})^3 \). Using the exponent rule \( (a^m)^n=a^{mn} \), we get \( 10^{\frac{1}{2}\times3}=10^{\frac{3}{2}} \).

Step9: Solve problem 8: \( \sqrt[6]{2} \) to exponential form

Using \( \sqrt[n]{a^m}=a^{\frac{m}{n}} \), here \( a = 2 \), \( m = 1 \), \( n = 6 \), so \( \sqrt[6]{2}=2^{\frac{1}{6}} \).

Step10: Solve problem 9: \( (\sqrt[4]{2})^5 \) to exponential form

\( \sqrt[4]{2}=2^{\frac{1}{4}} \), so \( (\sqrt[4]{2})^5=(2^{\frac{1}{4}})^5 \). Using \( (a^m)^n=a^{mn} \), we get \( 2^{\frac{1}{4}\times5}=2^{\frac{5}{4}} \).

Step11: Solve problem 10: \( (\sqrt[4]{5})^5 \) to exponential form

\( \sqrt[4]{5}=5^{\frac{1}{4}} \), so \( (\sqrt[4]{5})^5=(5^{\frac{1}{4}})^5 \). Using \( (a^m)^n=a^{mn} \), we get \( 5^{\frac{1}{4}\times5}=5^{\frac{5}{4}} \).

Step12: Solve problem 11: \( \sqrt[3]{2} \) to exponential form

\( \sqrt[3]{2}=2^{\frac{1}{3}} \).

Step13: Solve problem 12: \( \sqrt[6]{10} \) to exponential form

\( \sqrt[6]{10}=10^{\frac{1}{6}} \).

(Note: Some of the handwritten parts like \( \sqrt{7^{-2}} \), \( \sqrt[4]{4^{-3}} \) etc. might be typos or additional markings, but the core conversion is as above.)

Answer:

For each problem:

1. \( 7^{\frac{1}{2}} \) in radical form: \( \boldsymbol{\sqrt{7}} \)
2. \( 4^{\frac{4}{3}} \) in radical form: \( \boldsymbol{\sqrt[3]{4^4}} \)
3. \( 2^{\frac{5}{3}} \) in radical form: \( \boldsymbol{\sqrt[3]{2^5}} \)
4. \( 7^{\frac{4}{3}} \) in radical form: \( \boldsymbol{\sqrt[3]{7^4}} \)
5. \( 6^{\frac{3}{2}} \) in radical form: \( \boldsymbol{\sqrt{6^3}} \) (or \( (\sqrt{6})^3 \))
6. \( 2^{\frac{1}{6}} \) in radical form: \( \boldsymbol{\sqrt[6]{2}} \)
7. \( (\sqrt{10})^3 \) in exponential form: \( \boldsymbol{10^{\frac{3}{2}}} \)
8. \( \sqrt[6]{2} \) in exponential form: \( \boldsymbol{2^{\frac{1}{6}}} \)
9. \( (\sqrt[4]{2})^5 \) in exponential form: \( \boldsymbol{2^{\frac{5}{4}}} \)
10. \( (\sqrt[4]{5})^5 \) in exponential form: \( \boldsymbol{5^{\frac{5}{4}}} \)
11. \( \sqrt[3]{2} \) in exponential form: \( \boldsymbol{2^{\frac{1}{3}}} \)
12. \( \sqrt[6]{10} \) in exponential form: \( \boldsymbol{10^{\frac{1}{6}}} \)