If \(\rm \sqrt{1225 \times \sqrt{32 \div x}}= 70\) find the value of x.
2
Solving the Equation to Find the Value of x
The problem asks us to find the value of the variable 'x' from the given equation: \(\rm \sqrt{1225 \times \sqrt{32 \div x}}= 70\).
To solve this equation, we need to isolate 'x' by performing inverse operations step-by-step. Let's break down the process.
Step-by-Step Solution for Finding x
We start with the given equation:
\(\rm \sqrt{1225 \times \sqrt{32 \div x}}= 70\)
The entire left side of the equation is under a square root. To remove this outer square root, we need to square both sides of the equation.
\(\rm (\sqrt{1225 \times \sqrt{32 \div x}})^2 = 70^2\)
Squaring the square root cancels it out on the left side, and we calculate \(70^2\) on the right side:
\(\rm 1225 \times \sqrt{32 \div x} = 4900\)
Now, we have the term \(\rm \sqrt{32 \div x}\) multiplied by 1225. To isolate the square root term, we divide both sides of the equation by 1225.
\(\rm \sqrt{32 \div x} = \frac{4900}{1225}\)
Let's simplify the division on the right side:
\(\rm \frac{4900}{1225} = \frac{4 \times 1225}{1 \times 1225} = 4\)
So, the equation simplifies to:
\(\rm \sqrt{32 \div x} = 4\)
We now have another square root involving 'x'. To remove this inner square root, we square both sides of the equation again.
\(\rm (\sqrt{32 \div x})^2 = 4^2\)
Squaring the square root on the left side cancels it out, and we calculate \(4^2\) on the right side:
\(\rm 32 \div x = 16\)
This equation can be written as a fraction:
\(\rm \frac{32}{x} = 16\)
To solve for 'x', we can multiply both sides by 'x' to get 'x' out of the denominator:
\(\rm 32 = 16x\)
Finally, to find the value of 'x', we divide both sides by 16:
\(\rm x = \frac{32}{16}\)
\(\rm x = 2\)
Thus, the value of x that satisfies the given equation is 2.
Verification of the Solution
Let's substitute \(x = 2\) back into the original equation to verify the result:
\(\rm \sqrt{1225 \times \sqrt{32 \div x}} = 70\)
Substitute \(x = 2\):
\(\rm \sqrt{1225 \times \sqrt{32 \div 2}}\)
Calculate the division inside the inner square root:
\(\rm \sqrt{1225 \times \sqrt{16}}\)
Calculate the inner square root:
\(\rm \sqrt{1225 \times 4}\)
Calculate the product inside the outer square root:
\(\rm \sqrt{4900}\)
Calculate the final square root:
\(\rm \sqrt{4900} = \sqrt{70^2} = 70\)
Since the left side equals the right side (70 = 70), the value \(x=2\) is correct.
Based on our calculations, the value of x is 2.
Revision Table: Key Steps in Solving Radical Equations
| Step | Description | Applied Example (\(\rm \sqrt{A \times \sqrt{B/x}} = C\)) |
|---|---|---|
| 1 | Isolate the outermost radical if possible. (Not strictly necessary here, as it covers the entire expression) | \(\rm \sqrt{1225 \times \sqrt{32 \div x}}= 70\) (Already isolated) |
| 2 | Square both sides to eliminate the outer radical. | \(\rm 1225 \times \sqrt{32 \div x}= 4900\) |
| 3 | Isolate the next radical term. | \(\rm \sqrt{32 \div x}= \frac{4900}{1225} = 4\) |
| 4 | Square both sides again to eliminate the inner radical. | \(\rm 32 \div x= 16\) |
| 5 | Solve the resulting linear equation for x. | \(\rm x = \frac{32}{16} = 2\) |
| 6 | Verification: Substitute the value of x back into the original equation. | \(\rm \sqrt{1225 \times \sqrt{32 \div 2}} = \sqrt{1225 \times \sqrt{16}} = \sqrt{1225 \times 4} = \sqrt{4900} = 70\). Verified. |
Additional Information on Solving Equations with Square Roots
Equations containing square roots (or other radical expressions) are called radical equations. Solving them often involves squaring both sides of the equation one or more times to eliminate the radicals. It's crucial to remember the following:
- Squaring Property: Squaring both sides of an equation can sometimes introduce extraneous solutions. These are values that satisfy the squared equation but not the original equation. Therefore, it is always recommended to check your final answer by substituting it back into the original equation.
- Order of Operations: When an equation involves multiple terms or operations outside and inside the radical, use inverse operations (like addition/subtraction, multiplication/division) to isolate the radical term before squaring. In our problem, the \(\sqrt{1225}\) was implicitly multiplied with \(\sqrt{32 \div x}\) under the outer root, which is different from having them separated by a plus or minus sign.
- Simplifying Radicals: Sometimes, simplifying the square roots of known perfect squares (\(\sqrt{1225} = 35\)) at the beginning can make the equation easier to handle.
Understanding these concepts helps in confidently solving radical equations and verifying the correctness of the obtained value for the unknown variable like 'x'.
Important Questions from Simplification
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