Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Aug 26, 2023

Simplify the expression $\frac{y-33}{\sqrt{y}+\sqrt{33}}$ where all variables represent positive real numbers.

STEP 1

Assumptions1. All variable expressions represent positive real numbers. . The expression to simplify is $\frac{y-33}{\sqrt{y}+\sqrt{33}}$

STEP 2

We can simplify this expression by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator $\sqrt{y}+\sqrt{33}$ is $\sqrt{y}-\sqrt{33}$ .

STEP 3

Multiply the numerator and denominator by the conjugate of the denominator.
$\frac{y-33}{\sqrt{y}+\sqrt{33}} \times \frac{\sqrt{y}-\sqrt{33}}{\sqrt{y}-\sqrt{33}}$

STEP 4

Apply the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ , to the denominator.
$\frac{(y-33)(\sqrt{y}-\sqrt{33})}{y -33}$

STEP 5

Notice that the term $y -33$ cancels out in the numerator and denominator.
$\sqrt{y}-\sqrt{33}$ So, the simplified form of the given expression $\frac{y-33}{\sqrt{y}+\sqrt{33}}$ is $\sqrt{y}-\sqrt{33}$ .

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