Question
Rationalize the numerator of the number. Simplify if needed.
1+4√3 / 3√5
135√−43√
the fraction with numerator 1 and denominator 3 square root of 5 minus 4 square root of 3
4935√−1215√
the fraction with numerator 49 and denominator 3 square root of 5 minus 12 square root of 15
−4735√−1215√
the fraction with numerator negative 47 and denominator 3 square root of 5 minus 12 square root of 15
−4735√+1215√
(1 point)
1+4√3 / 3√5
135√−43√
the fraction with numerator 1 and denominator 3 square root of 5 minus 4 square root of 3
4935√−1215√
the fraction with numerator 49 and denominator 3 square root of 5 minus 12 square root of 15
−4735√−1215√
the fraction with numerator negative 47 and denominator 3 square root of 5 minus 12 square root of 15
−4735√+1215√
(1 point)
Answers
Answered by
GPT-4o mini
To rationalize the numerator of the expression \(\frac{1 + 4\sqrt{3}}{3\sqrt{5}}\), we will multiply both the numerator and the denominator by the conjugate of the numerator, which in this case is \(1 - 4\sqrt{3}\).
### Step 1: Identify the Expression
\[
\frac{1 + 4\sqrt{3}}{3\sqrt{5}}
\]
### Step 2: Multiply by the Conjugate
Multiply the numerator and denominator by the conjugate \(1 - 4\sqrt{3}\):
\[
\frac{(1 + 4\sqrt{3})(1 - 4\sqrt{3})}{(3\sqrt{5})(1 - 4\sqrt{3})}
\]
### Step 3: Simplify the Numerator
Using the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\):
\[
(1 + 4\sqrt{3})(1 - 4\sqrt{3}) = 1^2 - (4\sqrt{3})^2 = 1 - 16 \times 3 = 1 - 48 = -47
\]
### Step 4: Simplify the Denominator
Now, calculate the denominator:
\[
3\sqrt{5}(1 - 4\sqrt{3}) = 3\sqrt{5} - 12\sqrt{15}
\]
### Step 5: Combine the Results
The expression becomes:
\[
\frac{-47}{3\sqrt{5} - 12\sqrt{15}}
\]
### Step 6: Final Expression
Thus, the rationalized form is:
\[
\frac{-47}{3\sqrt{5} - 12\sqrt{15}}
\]
This matches up with one of the options you provided. The final result is:
\[
-\frac{47}{3\sqrt{5} - 12\sqrt{15}}
\]
This corresponds to the option:
\[
-\frac{47}{35\sqrt{} - 12\sqrt{15}}
\] where it looks like there may have been some simplification intended in your original options. If needed, please clarify which initial problem you'd like simplified further!
### Step 1: Identify the Expression
\[
\frac{1 + 4\sqrt{3}}{3\sqrt{5}}
\]
### Step 2: Multiply by the Conjugate
Multiply the numerator and denominator by the conjugate \(1 - 4\sqrt{3}\):
\[
\frac{(1 + 4\sqrt{3})(1 - 4\sqrt{3})}{(3\sqrt{5})(1 - 4\sqrt{3})}
\]
### Step 3: Simplify the Numerator
Using the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\):
\[
(1 + 4\sqrt{3})(1 - 4\sqrt{3}) = 1^2 - (4\sqrt{3})^2 = 1 - 16 \times 3 = 1 - 48 = -47
\]
### Step 4: Simplify the Denominator
Now, calculate the denominator:
\[
3\sqrt{5}(1 - 4\sqrt{3}) = 3\sqrt{5} - 12\sqrt{15}
\]
### Step 5: Combine the Results
The expression becomes:
\[
\frac{-47}{3\sqrt{5} - 12\sqrt{15}}
\]
### Step 6: Final Expression
Thus, the rationalized form is:
\[
\frac{-47}{3\sqrt{5} - 12\sqrt{15}}
\]
This matches up with one of the options you provided. The final result is:
\[
-\frac{47}{3\sqrt{5} - 12\sqrt{15}}
\]
This corresponds to the option:
\[
-\frac{47}{35\sqrt{} - 12\sqrt{15}}
\] where it looks like there may have been some simplification intended in your original options. If needed, please clarify which initial problem you'd like simplified further!