Simplify Algebraic Expression
The given expression is:
$ \sqrt{\frac{(3\frac{1}{4})^{4} - (4\frac{1}{3})^{4}}{(3\frac{1}{4})^{2} - (4\frac{1}{3})^{2}}} $Let $a = 3\frac{1}{4}$ and $b = 4\frac{1}{3}$. The expression inside the square root simplifies using the difference of squares formula, $x^2 - y^2 = (x-y)(x+y)$. Here, $x = a^2$ and $y = b^2$. So, $a^4 - b^4 = (a^2)^2 - (b^2)^2 = (a^2 - b^2)(a^2 + b^2)$.
The expression becomes:
$ \frac{(a^2 - b^2)(a^2 + b^2)}{a^2 - b^2} $Assuming $a^2 \neq b^2$, we can cancel the term $(a^2 - b^2)$, leaving $a^2 + b^2$. So, we need to find the square root of $a^2 + b^2$.
$ \sqrt{a^2 + b^2} $Convert Mixed Numbers to Fractions
Convert the mixed numbers $a$ and $b$ into improper fractions:
- $a = 3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4}$
- $b = 4\frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{13}{3}$
Calculate Squares and Sum
Calculate the squares of $a$ and $b$:
- $a^2 = (\frac{13}{4})^2 = \frac{169}{16}$
- $b^2 = (\frac{13}{3})^2 = \frac{169}{9}$
Now, add $a^2$ and $b^2$:
$ a^2 + b^2 = \frac{169}{16} + \frac{169}{9} $Find a common denominator, which is $16 \times 9 = 144$.
$ a^2 + b^2 = \frac{169 \times 9}{16 \times 9} + \frac{169 \times 16}{9 \times 16} = \frac{1521}{144} + \frac{2704}{144} $ $ a^2 + b^2 = \frac{1521 + 2704}{144} = \frac{4225}{144} $Find the Square Root
Calculate the square root of the sum $a^2 + b^2$:
$ \sqrt{\frac{4225}{144}} = \frac{\sqrt{4225}}{\sqrt{144}} $We know that $\sqrt{4225} = 65$ and $\sqrt{144} = 12$.
$ \frac{65}{12} $Convert to Mixed Number
Convert the improper fraction $\frac{65}{12}$ back to a mixed number:
$ \frac{65}{12} = 5 \text{ with a remainder of } 5 $So, the result is $5\frac{5}{12}$.
Similar Questions
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- Find the value of $57^{-8} \div 57^{17} \times 57^{-18}$.
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- Simplify: $(1^3 + 2^3 + 3^3 + \dots + 8^3)^{\frac{-5}{2}}$
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- Solve the following:
$(625)^{0.17} \times (625)^{0.08} = ?$ - If $\frac{9^m \times 3^5 \times 27^3}{3 \times 81^4} = 3^9$, then the value of m is:
- If $\sqrt{45} + \sqrt{125} = 17.88$ then what will be the value of $\sqrt{180} + \sqrt{80}$ ?
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Important Questions from Surds and Indices
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
The expression \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) is equal to:
Let \(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\) . Then √x is equal to:
If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
What is the value of \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
