12. Rationalize the denominator of the following: \( \frac{1}{\sqrt{5} + \sqrt{2}} \)
13. Simplify and find the value of:
(a) \( (243)^{\frac{6}{5}} \)
(b) \( (21)^{\frac{3}{2}} \times (21)^{\frac{5}{2}} \)
14. Given that \( \sqrt{10} = 3.162 \), find the value of \( \frac{1}{\sqrt{10}} \)
15. Find the product: \( \sqrt{2} \times \sqrt[4]{2} \times \sqrt[12]{32} \)
Answer (2)
The rationalized expression of 5 + 2 1 is 3 5 − 2 . The simplified values are 729 for ( 243 ) 5 6 , 194481 for ( 21 ) 2 3 × ( 21 ) 2 5 , approximately 0.316 for 10 1 , and 2 6 7 for the product 2 × 4 2 × 12 32 .
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Let's address each part of the question step-by-step:
Rationalize the denominator of 5 + 2 1 .
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 5 + 2 is 5 − 2 .
5 + 2 1 × 5 − 2 5 − 2 = ( 5 ) 2 − ( 2 ) 2 5 − 2
Calculating the denominator:
( 5 ) 2 − ( 2 ) 2 = 5 − 2 = 3
So the expression becomes:
3 5 − 2
Simplify and find the value of:
(a) ( 243 ) 5 6
First, rewrite 243 as a power of 3: 243 = 3 5 .
Therefore,
( 243 ) 5 6 = ( 3 5 ) 5 6 = 3 5 × 5 6 = 3 6
Calculating 3 6 :
3 6 = 729
(b) ( 21 ) 2 3 × ( 21 ) 2 5
When multiplying like bases, add the exponents:
( 21 ) 2 3 + 2 5 = ( 21 ) 4
Calculating ( 21 ) 4 :
2 1 4 = 21 × 21 × 21 × 21 = 194481
Given that 10 = 3.162 , find the value of 10 1 .
Using the given approximate value:
10 1 ≈ 3.162 1 ≈ 0.3162
Find the product: 2 × 4 2 × 12 32 .
First, rewrite each component using fractional exponents:
2 = 2 2 1 , 4 2 = 2 4 1 , 12 32 = ( 2 5 ) 12 1 = 2 12 5
Now, multiply the terms by adding the exponents:
2 2 1 + 4 1 + 12 5
Converting to a common denominator (12):
2 1 = 12 6 , 4 1 = 12 3 , 12 5 = 12 5
Add the exponents:
12 6 + 12 3 + 12 5 = 12 14 = 6 7
Thus, the product is:
2 6 7
The value of 2 6 7 can be further simplified or calculated as needed. However, this is the simplest algebraic form.
