Question

Blank radical 5 + 3 radical 45 equals 13 radical blank

Answer 1

To solve the equation \( x \sqrt{5} + 3 \sqrt{45} = 13 \sqrt{y} \), we first need to simplify \( 3 \sqrt{45} \).

Since \( \sqrt{45} = \sqrt{9 \cdot 5} = 3 \sqrt{5} \), we can simplify:

\[
3 \sqrt{45} = 3 \cdot 3 \sqrt{5} = 9 \sqrt{5}.
\]

Now we substitute this back into our equation:

\[
x \sqrt{5} + 9 \sqrt{5} = 13 \sqrt{y}.
\]

Combining like terms on the left side, we can factor out \( \sqrt{5} \):

\[
(x + 9) \sqrt{5} = 13 \sqrt{y}.
\]

For the equation to hold, the coefficients of \( \sqrt{5} \) and \( \sqrt{y} \) must match. Thus, we can write:

\[
x + 9 = 13 \quad \text{and} \quad \sqrt{y} = \sqrt{5}.
\]

Now, we can solve for \( x \):

\[
x + 9 = 13 \implies x = 13 - 9 = 4.
\]

Next, for \( \sqrt{y} = \sqrt{5} \):

\[
y = 5.
\]

Therefore, we have found the values of \( x \) and \( y \). The final answer is:

\[
\text{Blank radical } 5 + 3 \sqrt{45} = 13 \sqrt{5}.
\]

So, the value to fill in the blanks is \( 4 \) and \( 5 \).