Simplify :81/3 × (16)1/3 / (32)-1/3
Solution:
Given, the expression is \(\frac{(8)^{\frac{1}{3}}\times (16)^{\frac{1}{3}}}{(32)^{\frac{-1}{3}}}\)
We have to simplify the expression.
\((8)^{\frac{1}{3}}=(2^{3})^{\frac{1}{3}}\)
We know \((a^{m})^{n}=a^{mn}\)
= \((2)^{3\times \frac{1}{3}}\)
= \((2)^{\frac{3}{3}}\)
\((16)^{\frac{1}{3}}=(2^{4})^{\frac{1}{3}}\)
We know \((a^{m})^{n}=a^{mn}\)
= \((2)^{4\times \frac{1}{3}}\)
= \((2)^{\frac{4}{3}}\)
So, \((8)^{\frac{1}{3}}\times (16)^{\frac{1}{3}}=(2)^{\frac{3}{3}}\times (2)^{\frac{4}{3}}\)
We know \(a^{m}\times a^{n}=a^{m+n}\)
= \((2)^{\frac{3+4}{3}}\)
= \((2)^{\frac{7}{3}}\)
\((32)^{\frac{-1}{3}}=(2^{5})^{\frac{-1}{3}}\)
We know \((a^{m})^{n}=a^{mn}\)
= \((2)^{5\times \frac{-1}{3}}\)
= \((2)^{\frac{-5}{3}}\)
Now, \(\frac{(8)^{\frac{1}{3}}\times (16)^{\frac{1}{3}}}{(32)^{\frac{-1}{3}}}\) = \(\frac{(2)^{\frac{7}{3}}}{(2)^{\frac{-5}{3}}}\)
We know \(\frac{a^{m}}{a^{n}}=a^{m-n}\)
= \((2)^{\frac{7}{3}-\frac{-5}{3}}\)
= \((2)^{\frac{7}{3}+\frac{5}{3}}\)
= \((2)^{\frac{7+5}{3}}\)
= \((2)^{\frac{12}{3}}\)
= (2)⁴
= 16
Therefore, the simplified value is 16.
✦ Try This: Simplify: \(\frac{(9)^{\frac{1}{3}}\times (49)^{\frac{1}{3}}}{(64)^{\frac{-1}{3}}}\)
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 1
NCERT Exemplar Class 9 Maths Exercise 1.3 Problem 14(vii)
Simplify :81/3 × (16)1/3 / (32)-1/3
Summary:
To multiply any two rational numbers, we multiply their numerators and their denominators separately and simplify the resultant fraction. The simplified form of the expression is 16
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