Use tableles of squares and reciprocal to evaluate;
10/√0.625+4/√164
To evaluate this expression, we can first simplify the square roots in the denominators.
√0.625 = √(5/8) = √5 / √8 = (1/2)√5
√164 = √(4 * 41) = √4 * √41 = 2√41
Now, we can substitute these values back into the expression:
10/(1/2)√5 + 4/2√41
Simplifying further:
10 * 2√5 / 1 + 4 / 2√41
20√5 + 4 / 2√41
Now, we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
(20√5 + 4 / 2√41) * (√41 / √41)
(20√5 + 4)√41 / 2 * 41
(20√5 + 4)√41 / 82
So, the simplified expression is (20√5 + 4)√41 / 82.
To evaluate the given expression, we need to simplify the square roots and calculate the reciprocals.
Step 1: Simplify the square roots.
√0.625 = √(5/8)
Since 0.625 is equivalent to 5/8, we can write the square root as √(5/8).
To simplify the square root, we can separate the numerator and denominator:
√(5/8) = √5/√8
Step 2: Simplify the square roots further.
√5 is an irrational number, so we can't simplify it any further.
√8 can be simplified by factoring out perfect squares:
√8 = √(4 × 2) = √4 × √2 = 2√2
So, we can rewrite the expression using the simplified square roots as follows:
10/√0.625 + 4/√164 = 10/(√5/√8) + 4/√164 = 10/(√5 / (2√2)) + 4/√164
Step 3: Simplify the expression further.
To divide by a fraction, we can multiply by its reciprocal:
10/(√5 / (2√2)) = 10 * (2√2 / √5)
√164 can be simplified by factoring out perfect squares:
√164 = √(4 × 41) = √4 × √41 = 2√41
So, we can rewrite the expression with the simplified square roots again:
10 * (2√2 / √5) + 4/√164 = 20√2 / √5 + 4/(2√41) = 20√2 / √5 + 2/√41
Step 4: Calculate the reciprocals.
Reciprocal of √5: 1/√5
Reciprocal of √41: 1/√41
So, the expression becomes:
20√2 / √5 + 2/√41
Step 5: Rationalize the denominators.
To rationalize the denominators, we need to multiply both the numerator and denominator by the conjugate of the respective square root expression:
20√2 / √5 + 2/√41
= (20√2 / √5) * (√5 / √5) + (2/√41) * (√41 / √41)
= 20√10 / 5 + 2√41 / 41
= 4√10 + 2√41 / 5
Therefore, the value of the expression 10/√0.625 + 4/√164 is 4√10 + 2√41 / 5.