Math Problem Statement

I do not get how to simplify this at all can u help pls im dying rn

I do not get how to simplify this at all can u help pls im dying rn

Solution

Let’s simplify the given expression step by step. The expression is:

32(22)223\frac{3^2 \cdot (2^2)^{-2}}{2^3}

Step 1: Simplify the powers.

  • 32=93^2 = 9
  • For (22)2(2^2)^{-2}, apply the exponent rule (am)n=amn(a^m)^n = a^{m \cdot n}, so: (22)2=22(2)=24(2^2)^{-2} = 2^{2 \cdot (-2)} = 2^{-4}

Now, the expression becomes: 92423\frac{9 \cdot 2^{-4}}{2^3}

Step 2: Combine the powers of 2.

The next step is to simplify the powers of 2. Use the rule aman=am+na^m \cdot a^n = a^{m+n}. We need to combine 242^{-4} and 232^3.

First, notice that 242^{-4} and 232^3 are in division, so apply the rule aman=amn\frac{a^m}{a^n} = a^{m-n}:

24÷23=243=272^{-4} \div 2^3 = 2^{-4-3} = 2^{-7}

So the entire expression simplifies to: 9279 \cdot 2^{-7}

Step 3: Final simplified expression.

Since there are no further simplifications, the final simplified form is:

927\frac{9}{2^7}

Step 4: Optional evaluation.

If you'd like to express it as a decimal or fraction:

27=128,so91282^7 = 128, \quad \text{so} \quad \frac{9}{128}

Thus, the simplified expression is either 9128\frac{9}{128} or as a decimal approximately 0.07030.0703.


Would you like more details or have any questions? Here are 5 related problems to practice:

  1. Simplify 43(32)234\frac{4^3 \cdot (3^2)^{-2}}{3^4}.
  2. What is 52(23)124\frac{5^2 \cdot (2^3)^{-1}}{2^4} simplified?
  3. Evaluate (72)1(73)2(7^2)^{-1} \cdot (7^3)^{-2}.
  4. Simplify 64(25)36224\frac{6^4 \cdot (2^5)^{-3}}{6^2 \cdot 2^{-4}}.
  5. What is the decimal value of 726\frac{7}{2^6}?

Tip: Always apply the exponent rules first when simplifying expressions with exponents!

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Math Problem Analysis

Mathematical Concepts

Exponents
Algebra

Formulas

(a^m)^n = a^{m \cdot n}
a^m \cdot a^n = a^{m+n}
\frac{a^m}{a^n} = a^{m-n}

Theorems

Exponent Rules

Suitable Grade Level

Grades 9-11