Class 9 Mathematics: Exponents and Radicals
Class 9th maths exercise 1.5
1. Evaluate the Following:
(i) \(64^{\frac{1}{2}}\)
Step 1: Recognize that \(a^{\frac{1}{n}} = \sqrt[n]{a}\).
Step 2: Apply to \(64^{\frac{1}{2}}\):
\[
64^{\frac{1}{2}} = \sqrt{64} = 8 \quad (\text{since } 8 \times 8 = 64)
\]
(ii) \(32^{\frac{1}{5}}\)
Step 1: Convert to radical form: \(\sqrt[5]{32}\).
Step 2: Simplify:
\[
32 = 2^5 \implies \sqrt[5]{2^5} = 2
\]
(iii) \(125^{\frac{1}{3}}\)
Step 1: Identify the cube root: \(\sqrt[3]{125}\).
Step 2: Calculate:
\[
125 = 5^3 \implies \sqrt[3]{5^3} = 5
\]
2. Simplify the Expressions:
(i) \(9^{\frac{3}{2}}\)
Step 1: Use \(a^{\frac{m}{n}} = (\sqrt[n]{a})^m\).
Step 2: Break down \(9^{\frac{3}{2}}\):
\[
(\sqrt{9})^3 = 3^3 = 27
\]
(ii) \(32^{\frac{2}{5}}\)
Step 1: Simplify the radical first: \(\sqrt[5]{32} = 2\).
Step 2: Raise to power 2:
\[
(2)^2 = 4
\]
(iii) \(16^{\frac{3}{4}}\)
Step 1: Find the 4th root: \(\sqrt[4]{16} = 2\).
Step 2: Cube the result:
\[
2^3 = 8
\]
(iv) \(125^{-\frac{1}{3}}\)
Step 1: Apply negative exponent rule: \(a^{-m} = \frac{1}{a^m}\).
Step 2: Simplify:
\[
125^{-\frac{1}{3}} = \frac{1}{125^{\frac{1}{3}}} = \frac{1}{5}
\]
3. Simplify Using Exponent Rules:
(i) \(2^{\frac{2}{3}} \cdot 2^{\frac{1}{5}}\)
Step 1: Use \(a^m \cdot a^n = a^{m+n}\).
Step 2: Add exponents:
\[
\frac{2}{3} + \frac{1}{5} = \frac{10 + 3}{15} = \frac{13}{15}
\]
Result: \(2^{\frac{13}{15}}\)
(ii) \(\left(\frac{1}{3}\right)^{\frac{7}{3}}\)
Step 1: Apply power to numerator and denominator:
\[
\frac{1^{\frac{7}{3}}}{3^{\frac{7}{3}}} = \frac{1}{3^{\frac{7}{3}}}
\]
Simplified Form: \(3^{-\frac{7}{3}}\)
(iii) \(\frac{11^{\frac{1}{2}}}{11^{\frac{1}{4}}}\)
Step 1: Use \(\frac{a^m}{a^n} = a^{m-n}\).
Step 2: Subtract exponents:
\[
\frac{1}{2} – \frac{1}{4} = \frac{1}{4}
\]
Result: \(11^{\frac{1}{4}}\)
(iv) \(7^{\frac{1}{2}} \cdot 8^{\frac{1}{2}}\)
Step 1: Combine using \((ab)^n = a^n \cdot b^n\):
\[
(7 \cdot 8)^{\frac{1}{2}} = 56^{\frac{1}{2}} = \sqrt{56}
\]
Simplified Radical: \(2\sqrt{14}\)
Key Exponent Rules
- \(a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\)
- \(a^{-m} = \frac{1}{a^m}\)
- \(a^m \cdot a^n = a^{m+n}\)
- \(\frac{a^m}{a^n} = a^{m-n}\)
- \((ab)^n = a^n \cdot b^n\)
Practice Problems
1. Simplify: \(81^{\frac{3}{4}}\)
Solution:
\[
81^{\frac{3}{4}} = (\sqrt[4]{81})^3 = 3^3 = 27
\]
2. Find: \(81^{\frac{1}{4}}\)
\(= \sqrt[4]{81} = 3\)
3. Simplify: \(5^{\frac{2}{3}} \cdot 5^{\frac{1}{6}}\)
\(= 5^{\frac{2}{3} + \frac{1}{6}}} = 5^{\frac{5}{6}}}\)
4. Simplify: \(\left(\frac{2}{3}\right)^{\frac{3}{2}}\)
\(= \frac{2^{\frac{3}{2}}}{3^{\frac{3}{2}}} = \frac{\sqrt{8}}{\sqrt{27}}}\)
5. Find: \(64^{-\frac{1}{3}}\)
\(= \frac{1}{64^{\frac{1}{3}}} = \frac{1}{4}\)
