Solve sqrt{(4.5)^2-(3/16)^2}

$$\sqrt { ( 4.5 ) ^ { 2 } - ( \frac { 3 } { 16 } ) ^ { 2 } }$$

Evaluate

$$\frac{15\sqrt{23}}{16}\approx 4.496092053$$

Solution Steps

Calculate 4.5 to the power of 2 and get 20.25.

$$\sqrt{20.25-\left(\frac{3}{16}\right)^{2}}$$

Calculate \frac{3}{16} to the power of 2 and get \frac{9}{256}.

$$\sqrt{20.25-\frac{9}{256}}$$

Convert decimal number 20.25 to fraction \frac{2025}{100}. Reduce the fraction \frac{2025}{100} to lowest terms by extracting and canceling out 25.

$$\sqrt{\frac{81}{4}-\frac{9}{256}}$$

Least common multiple of 4 and 256 is 256. Convert \frac{81}{4} and \frac{9}{256} to fractions with denominator 256.

$$\sqrt{\frac{5184}{256}-\frac{9}{256}}$$

Since \frac{5184}{256} and \frac{9}{256} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{5184-9}{256}}$$

Subtract 9 from 5184 to get 5175.

$$\sqrt{\frac{5175}{256}}$$

Rewrite the square root of the division \sqrt{\frac{5175}{256}} as the division of square roots \frac{\sqrt{5175}}{\sqrt{256}}.

$$\frac{\sqrt{5175}}{\sqrt{256}}$$

Factor 5175=15^{2}\times 23. Rewrite the square root of the product \sqrt{15^{2}\times 23} as the product of square roots \sqrt{15^{2}}\sqrt{23}. Take the square root of 15^{2}.

$$\frac{15\sqrt{23}}{\sqrt{256}}$$

Calculate the square root of 256 and get 16.

$$\frac{15\sqrt{23}}{16}$$