\frac{8 \sqrt{8}}{\sqrt{7}}
Answer (2)
Simplify the square root in the numerator: 8 = 2 2 .
Substitute the simplified square root back into the expression: 7 8 ( 2 2 ) = 7 16 2 .
Rationalize the denominator by multiplying both the numerator and denominator by 7 : 7 16 2 ⋅ 7 7 = 7 16 14 .
The simplified expression is 7 16 14 .
Explanation
Understanding the Problem We are given the expression 7 8 8 and our goal is to simplify it.
Simplifying the Numerator First, we simplify the square root in the numerator. We know that 8 = 4 ⋅ 2 , so 8 = 4 ⋅ 2 = 4 ⋅ 2 = 2 2 .
Substituting Back Now we substitute this back into the original expression: 7 8 8 = 7 8 ( 2 2 ) = 7 16 2
Rationalizing the Denominator To rationalize the denominator, we multiply both the numerator and the denominator by 7 : 7 16 2 ⋅ 7 7 = 7 16 2 7 = 7 16 14
Final Result Therefore, the simplified expression is 7 16 14 .
Examples
In engineering, you might need to calculate the stress on a material. If the stress is proportional to 7 14 , knowing how to simplify expressions like 7 8 8 helps in determining the stress value more accurately. This skill is also useful in physics when dealing with wave equations or energy calculations where square roots and rationalization are common.
To simplify 7 8 8 , we first simplify the square root in the numerator to get 7 16 2 and then rationalize the denominator. The final result is 7 16 14 .
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