Simplifying Square Roots

The question asks us to calculate the value of $\sqrt{180} + \sqrt{80}$, given the value of $\sqrt{45} + \sqrt{125}$. The final answer should be rounded to one decimal place.

Key Term Simplification

First, simplify all the square root terms by factoring out perfect squares:

• $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$
• $\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}$
• $\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}$
• $\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}$

Using Given Equation

We are given that $\sqrt{45} + \sqrt{125} = 17.89$. Substitute the simplified terms:

$3\sqrt{5} + 5\sqrt{5} = 17.89$

Combine the terms: $8\sqrt{5} = 17.89$.

Calculating Target Value

We need to find the value of $\sqrt{180} + \sqrt{80}$. Substitute the simplified terms:

$\sqrt{180} + \sqrt{80} = 6\sqrt{5} + 4\sqrt{5}$

Combine the terms: $10\sqrt{5}$.

Finding the Final Value

We know $8\sqrt{5} = 17.89$. We need to find $10\sqrt{5}$.

We can find the value of $\sqrt{5}$ by dividing the given value by 8: $\sqrt{5} = \frac{17.89}{8}$.

Now substitute this into the expression we need to find:

$10\sqrt{5} = 10 \times \frac{17.89}{8}$

$10\sqrt{5} = \frac{10}{8} \times 17.89 = \frac{5}{4} \times 17.89$

$10\sqrt{5} = 1.25 \times 17.89 = 22.3625$

Rounding the Result

The question asks for the answer to one decimal place. Rounding $22.3625$ to one decimal place gives $22.4$.