Simplify Exponential Expressions and Fractional Exponents

Math Problem Statement

If x = (2^8 * 3^15 * 24^5 * 15^3 * 10^8) / (9^20 * 12^12 * 20^10), then find the simplest form of x. Simplify: [(4096)^(1/4) + (4913)^(1/3)]^(3/4).

Let's address the two questions step by step:

The formula provided is: $x = \frac{2^8 \cdot 3^{15} \cdot 24^5 \cdot 15^3 \cdot 10^8}{9^{20} \cdot 12^{12} \cdot 20^{10}}$

Rewrite all terms using their prime factorizations:
- $24 = 2^3 \cdot 3$
- $15 = 3 \cdot 5$
- $10 = 2 \cdot 5$
- $9 = 3^2$
- $12 = 2^2 \cdot 3$
- $20 = 2^2 \cdot 5$
Substituting these factorizations: $x = \frac{2^8 \cdot 3^{15} \cdot (2^3 \cdot 3)^5 \cdot (3 \cdot 5)^3 \cdot (2 \cdot 5)^8}{(3^2)^{20} \cdot (2^2 \cdot 3)^{12} \cdot (2^2 \cdot 5)^{10}}$

Numerator: $= 2^{8+15+8} \cdot 3^{15+5+3} \cdot 5^{3+8} = 2^{31} \cdot 3^{23} \cdot 5^{11}$$$
Denominator: $= 3^{40} \cdot (2^{24} \cdot 3^{12}) \cdot (2^{20} \cdot 5^{10}) = 2^{24+20} \cdot 3^{40+12} \cdot 5^{10} = 2^{44} \cdot 3^{52} \cdot 5^{10}$$$

= 2^{31-44} \cdot 3^{23-52} \cdot 5^{11-10} = 2^{-13} \cdot 3^{-29} \cdot 5^1$$ Rewrite with positive exponents: $$x = \frac{5}{2^{13} \cdot 3^{29}}$$ --- ### 2. Simplify $$\left[(4096)^{\frac{1}{4}} + (4913)^{\frac{1}{3}}\right]^{\frac{3}{4}}$$: #### Break into steps: - $$(4096)^{\frac{1}{4}} = \sqrt[4]{4096} = 6$$ (since $$6^4 = 4096$$). - $$(4913)^{\frac{1}{3}} = \sqrt[3]{4913} = 17$$ (since $$17^3 = 4913$$). Simplify inside the brackets: $$(4096)^{\frac{1}{4}} + (4913)^{\frac{1}{3}} = 6 + 17 = 23$$ Raise to $$\frac{3}{4}$$: $$\left[23\right]^{\frac{3}{4}}$$ If you need a decimal value for $$23^{3/4}$$, it can be approximated numerically. --- Would you like detailed numeric approximations or further steps? Here are some relative questions for your practice: 1. How can exponents with negative values be simplified further? 2. What techniques are useful for simplifying large exponents in fractions? 3. How does rewriting terms into prime factorizations simplify the problem? 4. What are practical uses of fractional exponents in real-world applications? 5. Can $$\left[23^{3/4}\right]$$ be expressed in terms of a logarithmic function? **Tip:** Always simplify powers and roots step by step to avoid errors in large numbers!

Drop file here or Click Here to upload

Simplification of exponential expressions
Fractional exponents
Prime factorization

a^m * a^n = a^(m+n)
a^(m/n) = n-th root of (a^m)
(a^m)/(a^n) = a^(m-n)

Properties of exponents
Prime factorization techniques

Grades 9-12

Simplifying Expression with Exponents: 3^6 × (24)^(-2) over 18^3 × (2^(-3))^4

Simplifying Exponential Expressions: 5^3 + 4^3 * 4^7 - (12^10 / 3^10)

Simplifying Expressions Involving Exponents and Fractions

Simplifying Exponential Expressions with Prime Factorization

Solve Exponentiation Problem with Detailed Steps - 7^8 / 7^9 * 8^10