Verified Answer :
- Rewrite the expression using the rule for negative exponents: \(4^{-\frac{9}{2}} = \frac{1}{4^{\frac{9}{2}}}\).
- Express the base 4 as \(2^2\) to use the property \((2^2)^{\frac{9}{2}} = 2^{2 \times \frac{9}{2}} = 2^9\).
- Calculate the value of \(2^9 = 512\).
- State the final result: \(\boxed{\frac{1}{512}}\).
Explanation
- Analyzing the Problem and Applying the Reciprocal Rule
We are tasked with evaluating the expression \(4^{-\frac{9}{2}}\). This expression involves a constant base (4) raised to a negative rational exponent. To simplify this, we can rely on standard rules of exponents. Specifically, we will use the property that a negative exponent signifies the reciprocal: \(a^{-n} = \frac{1}{a^n}\), where \(a\) is a non-zero base and \(n\) is a positive exponent. Thus, we can rewrite the expression as \[\frac{1}{4^{\frac{9}{2}}}.\]
- Simplifying the Base and Exponent
Next, we notice that the base 4 is a perfect square, specifically \(2^2\). Substituting this into our expression, we have: \[\frac{1}{(2^2)^{\frac{9}{2}}}.\] Applying the power rule for exponents, which states that \((a^m)^n = a^{m \cdot n}\), we can multiply the exponents: \(2 \times \frac{9}{2} = 9\). Therefore, the denominator simplifies to \(2^9\).
- Calculating the Final Value
Now, we evaluate the power of 2 in the denominator: \(2^9 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\). Multiplying these factors together: \(2 \times 2 = 4\), \(4 \times 2 = 8\), \(8 \times 2 = 16\), \(16 \times 2 = 32\), \(32 \times 2 = 64\), \(64 \times 2 = 128\), \(128 \times 2 = 256\), \(256 \times 2 = 512\). Thus, \(2^9 = 512\). Substituting this back into our fraction, we get \(\frac{1}{512}\).
Examples
Negative exponents are frequently used in scientific notation and engineering to express extremely small values. For instance, if a tiny particle has a mass of \(4^{-\frac{9}{2}}\) grams, we can easily determine that it weighs \(\frac{1}{512}\) of a gram. This simplifies calculations when dealing with physics problems where large and small numbers coexist, allowing researchers to keep numbers manageable by expressing them as powers of a base.