simplify the following
(4a^-2)^1/2
simplify the following
(sqroot of 2 - sqroot of 7)/(sqroot of 2 + sqroot of 7)
To solve (4a^-2)^(1/2), we can apply the exponent rule which states that when raising a power to another power, we multiply the exponents. So, we have:
(4a^-2)^(1/2) = 4^(1/2) * (a^-2)^(1/2)
Now, we simplify the exponents:
4^(1/2) = √4 = 2
(a^-2)^(1/2) = a^(-2/2) = a^-1
Substituting these simplified exponents back into the equation, we have:
(4a^-2)^(1/2) = 2 * a^-1
Since a^-1 is equivalent to 1/a, then the final simplified form is:
(4a^-2)^(1/2) = 2 * 1/a = 2/a
Next, to simplify (√2 - √7)/(√2 + √7), we can multiply both the numerator and the denominator by the conjugate of the denominator, which is (√2 - √7). Applying this operation, we get:
[(√2 - √7) * (√2 - √7)] / [(√2 + √7) * (√2 - √7)]
Simplifying the numerator and denominator using the distributive property, we have:
[2 - 2√7 - 2√7 + 7] / [√2 * √2 - √2 * √7 + √7 * √2 - √7 * √7]
Simplifying further, we have:
[9 - 4√7] / [2 - √14 - √14 + 7]
Combining like terms in the denominator, we get:
[9 - 4√7] / [9 - 2√14]
Therefore, the simplified form of (√2 - √7)/(√2 + √7) is:
[9 - 4√7] / [9 - 2√14]