Given \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\). Which one of the following statements is TRUE?
9 inches = 0.25 yards
The question asks us to identify the true statement given the equation \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\). This equation involves fractional exponents, specifically the power of \(\frac{1}{2}\), which represents a square root.
Equation Analysis: Inches and Yards
The given equation is:
\[{\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\]
We know that an exponent of \(\frac{1}{2}\) is equivalent to taking the square root. So, the equation can be conceptually understood as:
\[\sqrt{9{\rm{\;inches}}} = \sqrt{0.25{\rm{\;yards}}}\]
To eliminate the fractional exponents and simplify the equation to a direct comparison like "X inches = Y yards", we can square both sides of the original equation. Squaring an expression raised to the power of \(\frac{1}{2}\) will remove the exponent, as \({\left( {a^{\frac{1}{2}}} \right)^2} = a^{\frac{1}{2} \times 2} = a^1 = a\).
Solution Steps: Inches and Yards Equation
Let's square both sides of the given equation:
\[{\left( {{\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}}} \right)^2} = {\left( {{\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}} \right)^2}\]
Applying the exponent rule \({\left( {a^m} \right)^n} = a^{m \times n}\), we multiply the exponents:
- For the left side: \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2} \times 2}} = {\left( {9{\rm{\;inches}}} \right)^1} = 9{\rm{\;inches}}\)
- For the right side: \({\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2} \times 2}} = {\left( {0.25{\rm{\;yards}}} \right)^1} = 0.25{\rm{\;yards}}\)
Therefore, after squaring both sides, the equation simplifies to a clear equality:
\[9{\rm{\;inches}} = 0.25{\rm{\;yards}}\]
Statement Comparison: Finding the True Statement
Now, let's compare our derived statement with the given options to find which one is TRUE.
| Option | Statement | Analysis |
|---|---|---|
| 1 | \(3{\rm{\;inches}} = 0.5{\rm{\;yards}}\) | This statement would be obtained if we took the square root of the numerical values (i.e., \(\sqrt{9} = 3\) and \(\sqrt{0.25} = 0.5\)). However, the given equation already expresses the equality of the square roots of the total expressions. To find the direct relationship without the exponent, we square both sides of the original equation. |
| 2 | \(9{\rm{\;inches}} = 1.5{\rm{\;yards}}\) | This statement does not match our derived result of \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\). |
| 3 | \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\) | This statement exactly matches our derived result from simplifying the given equation. This is the true statement. |
| 4 | \(81{\rm{\;inches}} = 0.0625{\rm{\;yards}}\) | This statement would be obtained by squaring both sides of the true statement \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\) (i.e., \(9^2 = 81\) and \(0.25^2 = 0.0625\)). The question asks for the statement directly implied by the *given* equation, not a further transformation of its simplified form. |
From the comparison, it is clear that the statement \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\) is the true statement derived directly from the given equation.
Important Questions from Numerical Estimation
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