If x = \(\sqrt{64}+ \sqrt{121} - \sqrt{169}\), then find the value of x2.
Understanding the Square Root Problem
The problem asks us to first calculate the value of \(x\) based on a given expression involving square roots, and then find the value of \(x^2\).
The expression for \(x\) is given as: \[x = \sqrt{64} + \sqrt{121} - \sqrt{169}\]
To find \(x\), we need to calculate the value of each square root term in the expression.
Calculating the Square Root Terms
Let's find the value of each square root:
- The square root of 64: \(\sqrt{64}\)
- The square root of 121: \(\sqrt{121}\)
- The square root of 169: \(\sqrt{169}\)
We know that the square root of a number is a value that, when multiplied by itself, gives the original number.
- For \(\sqrt{64}\), we think of a number multiplied by itself that equals 64. That number is 8, because \(8 \times 8 = 64\). So, \(\sqrt{64} = 8\).
- For \(\sqrt{121}\), we think of a number multiplied by itself that equals 121. That number is 11, because \(11 \times 11 = 121\). So, \(\sqrt{121} = 11\).
- For \(\sqrt{169}\), we think of a number multiplied by itself that equals 169. That number is 13, because \(13 \times 13 = 169\). So, \(\sqrt{169} = 13\).
Substituting Values and Solving for x
Now we substitute these calculated square root values back into the expression for \(x\):
\[x = \sqrt{64} + \sqrt{121} - \sqrt{169}\] \[x = 8 + 11 - 13\]Next, we perform the addition and subtraction:
\[x = (8 + 11) - 13\] \[x = 19 - 13\] \[x = 6\]So, the value of \(x\) is 6.
Finding the Value of x Squared (x²)
The question asks for the value of \(x^2\). Now that we have found \(x = 6\), we can calculate \(x^2\):
\[x^2 = 6^2\]Calculating \(6^2\) means multiplying 6 by itself:
\[x^2 = 6 \times 6\] \[x^2 = 36\]Therefore, the value of \(x^2\) is 36.
Summary of Steps
| Step | Calculation | Result |
|---|---|---|
| 1: Calculate \(\sqrt{64}\) | \(\sqrt{64}\) | 8 |
| 2: Calculate \(\sqrt{121}\) | \(\sqrt{121}\) | 11 |
| 3: Calculate \(\sqrt{169}\) | \(\sqrt{169}\) | 13 |
| 4: Substitute into \(x\) expression | \(x = 8 + 11 - 13\) | \(x = 6\) |
| 5: Calculate \(x^2\) | \(x^2 = 6^2\) | 36 |
The final value of \(x^2\) is 36.
Revision Table: Square Root Values
| Number | Square Root |
|---|---|
| 64 | 8 |
| 121 | 11 |
| 169 | 13 |
Remembering common perfect squares and their roots is helpful for solving these types of problems quickly.
Additional Information: Understanding Square Roots and Exponents
Square Root (\(\sqrt{}\)): The square root of a non-negative number \(N\) is a number \(r\) such that \(r \times r = N\). It's the inverse operation of squaring a number.
Exponents (\(x^2\)): An exponent indicates how many times a base number is multiplied by itself. In \(x^2\), \(x\) is the base and 2 is the exponent, meaning \(x\) is multiplied by itself two times (\(x \times x\)).
In this problem, we combined the operations of finding square roots and then squaring the result. The order of operations is important: first find \(x\) by performing the square root calculations and then the addition/subtraction, and finally square the value of \(x\).
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