Question

1. use exponent properties to find the value of \\(\frac{5^{18}3^{4}(2^{3})^{3}}{2^{10}3^{2}25^{9}}\\) without using a calculator.

Explanation:

Step1: Simplify numerator exponents

First, handle the numerator: \(5^{18} \times 3^{4} \times (2^{3})^{3}\). Using the power of a power property \((a^m)^n = a^{m \times n}\), \((2^{3})^{3}=2^{3\times3}=2^{9}\). So the numerator becomes \(5^{18} \times 3^{4} \times 2^{9}\).

Step2: Simplify denominator exponents

Now the denominator: \(2^{10} \times 3^{2} \times 25^{9}\). Note that \(25 = 5^{2}\), so \(25^{9}=(5^{2})^{9}\). Using the power of a power property, \((5^{2})^{9}=5^{2\times9}=5^{18}\). So the denominator becomes \(2^{10} \times 3^{2} \times 5^{18}\).

Step3: Rewrite the fraction

Now the expression is \(\frac{5^{18} \times 3^{4} \times 2^{9}}{2^{10} \times 3^{2} \times 5^{18}}\).

Step4: Cancel common bases

Cancel out the common bases. For \(5\): \(5^{18}/5^{18} = 1\). For \(2\): Using the quotient of powers property \(a^m / a^n = a^{m - n}\), \(2^{9}/2^{10}=2^{9 - 10}=2^{-1}=\frac{1}{2}\). For \(3\): \(3^{4}/3^{2}=3^{4 - 2}=3^{2}=9\).

Step5: Multiply the results

Multiply the remaining terms: \(1\times9\times\frac{1}{2}=\frac{9}{2}\). Wait, that seems off. Wait, let's re - check. Wait, numerator: \(5^{18}\times3^{4}\times2^{9}\), denominator: \(2^{10}\times3^{2}\times5^{18}\). So \(5^{18}\) cancels, \(3^{4}/3^{2}=3^{2} = 9\), \(2^{9}/2^{10}=2^{-1}=\frac{1}{2}\). Then \(9\times\frac{1}{2}=\frac{9}{2}\)? Wait, maybe I misread the original problem. Wait, the original problem: Let's re - parse the original expression. The user wrote: \(\frac{5^{18}3^{4}(2^{3})^{3}}{2^{10}3^{2}25^{9}}\). Wait, \(25^{9}=(5^{2})^{9}=5^{18}\), so numerator has \(5^{18}\), denominator has \(5^{18}\), so they cancel. Numerator has \(3^{4}\), denominator has \(3^{2}\), so \(3^{4 - 2}=3^{2}=9\). Numerator has \(2^{9}\), denominator has \(2^{10}\), so \(2^{9-10}=2^{-1}=\frac{1}{2}\). Then \(9\times\frac{1}{2}=\frac{9}{2}\)? Wait, but maybe the original problem was written differently. Wait, maybe the denominator is \(2^{10}*3^{2}*25^{9}\) and numerator is \(5^{18}3^{4}(2^{3})^{3}\). Yes, so the calculation is correct. Wait, but let's check again.

Wait, \((2^{3})^{3}=2^{9}\), correct. \(25^{9}=(5^{2})^{9}=5^{18}\), correct. So the fraction is \(\frac{5^{18}\times3^{4}\times2^{9}}{2^{10}\times3^{2}\times5^{18}}\). Cancel \(5^{18}\), we get \(\frac{3^{4}\times2^{9}}{2^{10}\times3^{2}}\). Then \(\frac{3^{4}}{3^{2}} = 3^{2}=9\), \(\frac{2^{9}}{2^{10}}=\frac{1}{2}\). Then \(9\times\frac{1}{2}=\frac{9}{2}\).

Answer:

\(\frac{9}{2}\)