So I am supposed to solve this without using a calculator: Sqrt[20]/10 - Sqrt[10]/Sqrt[32] - Sqrt[0.3125] + Sqrt[3 + 1/5]
You can put this into WolframAlpha as is to make it prettier.
Answer given is 1/2 * SQRT(5)
I really don't know where to start here. I have 5 sqrt rules in front of me along with 5 power rules.
Interpretation:
√20/10 - √10/√32 - √.3125 + √(3+1/5)
= 2√5/10 - √10/√32 * √32/√32 - √(5/16) + √(16/5)
= 2√5/10 - 8√5/32 - √5/4 + 4/√5 *√5/√5
= √5/5 - √5/4 - √5/4 + 4√5/5
= √5(1/5 - 1/4 - 1/4 + 4/5)
= (1/2)√5 or √5/2
Thank you Reiny, much appreciated!
To simplify the given expression without using a calculator, you can try to simplify each square root expression separately. Let's break it down step by step:
1. Simplifying the first square root expression:
Recall the square root rule that √(a * b) = √a * √b.
√20 can be written as √(4 * 5), which simplifies to 2 * √5.
Therefore, the first term becomes (2 * √5) / 10 = √5 / 5.
2. Simplifying the second square root expression:
The second term √10 / √32 can be simplified by rationalizing the denominator.
Rationalizing the denominator of √32, we get √(32 * 1/32) = √32 / √(32 * 32), which simplifies to 1 / 4.
So, the second term becomes √10 * 1/4 = √10 / 4.
3. Simplifying the third square root expression:
√0.3125 can be written as √(5/16), which simplifies to 1/4.
Therefore, the third term becomes 1/4.
4. Simplifying the fourth square root expression:
√(3 + 1/5) can be simplified as √(16/5), which simplifies to 2/√5.
So, the fourth term becomes 2/√5.
Now, adding all the simplified terms together:
√5 / 5 - √10 / 4 - 1/4 + 2/√5
To combine the terms with square roots, we need to have like denominators.
To do this, multiply the fraction with √5 in the numerator and denominator of the first term, and the fraction with 4 in the numerator and denominator of the second term:
(√5 * √5) / (5*√5) - √10 * 4 / (4*√10) - 1/4 + 2/√5
Simplifying the expression further:
5 / (5*√5) - 4√10 / (4*√10) - 1/4 + 2/√5
Cancel out the common factors:
1/√5 - √10 / √10 - 1/4 + 2/√5
Simplifying further:
1/√5 - 1 - 1/4 + 2/√5
Now, let's combine the terms without square roots:
-1 - 1/4 = -5/4
And the terms with square roots:
1/√5 + 2/√5 = 3/√5
Finally, combining both parts:
-5/4 + 3/√5
To rationalize the denominator, multiply the fraction by √5/√5:
(-5/4) * (√5/√5) + (3/√5) * (√5/√5)
-5√5 / (4√5) + 3√5 / (√5 * √5)
Simplifying:
-5√5 / (4√5) + 3√5 / 5
Combining:
(-5√5 + 3√5) / (4√5)
Simplifying further:
-2√5 / (4√5)
Canceling the common factors:
-1/2
So, after simplifying the given expression, the answer becomes -1/2, not 1/2√5 as stated in the question. Please double-check the initial problem statement and verify it again.