Math Problem Statement

(49m².n²)³ : (7m⁸.n)⁵

Solution

We aim to simplify the expression:

(49m2n2)3(7m8n)5.\frac{(49m^2n^2)^3}{(7m^8n)^5}.

Step 1: Expand the powers of each term.

For the numerator (49m2n2)3(49m^2n^2)^3: 493(m2)3(n2)3=493m6n6.49^3 \cdot (m^2)^3 \cdot (n^2)^3 = 49^3 \cdot m^6 \cdot n^6.

For the denominator (7m8n)5(7m^8n)^5: 75(m8)5n5=75m40n5.7^5 \cdot (m^8)^5 \cdot n^5 = 7^5 \cdot m^{40} \cdot n^5.

Step 2: Write the expanded form.

The expression becomes: 493m6n675m40n5.\frac{49^3 \cdot m^6 \cdot n^6}{7^5 \cdot m^{40} \cdot n^5}.

Step 3: Simplify coefficients.

Since 49=7249 = 7^2, 493=(72)3=7649^3 = (7^2)^3 = 7^6. Substituting this: 76m6n675m40n5.\frac{7^6 \cdot m^6 \cdot n^6}{7^5 \cdot m^{40} \cdot n^5}.

Now, simplify 76/75=765=71=77^6 / 7^5 = 7^{6-5} = 7^1 = 7.

Step 4: Simplify the powers of mm and nn.

For mm, m6/m40=m640=m34m^6 / m^{40} = m^{6-40} = m^{-34}.
For nn, n6/n5=n65=n1=nn^6 / n^5 = n^{6-5} = n^1 = n.

Step 5: Combine results.

The simplified expression is: 7nm34=7nm34.\frac{7 \cdot n}{m^{34}} = \frac{7n}{m^{34}}.


Final Answer:

7nm34\frac{7n}{m^{34}}


Would you like further clarification or examples? Here are some related questions for practice:

  1. Simplify (16x3y2)2/(4x5y4)(16x^3y^2)^2 / (4x^5y^4).
  2. Expand and simplify (2a2b)4(ab3)2(2a^2b)^4 \cdot (ab^3)^2.
  3. Simplify (25x6)1/25x2\frac{(25x^6)^{1/2}}{5x^2}.
  4. If m=2m = 2 and n=3n = 3, calculate the value of 7nm34\frac{7n}{m^{34}}.
  5. How does the law of exponents simplify calculations with fractional bases?

Tip: Always simplify coefficients before handling exponents for more straightforward calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Exponent Laws
Simplification of Rational Expressions

Formulas

Power of a product: (ab)^n = a^n * b^n
Law of exponents: a^m / a^n = a^(m-n)
Simplification of rational expressions

Theorems

Exponentiation rules
Simplification of rational expressions

Suitable Grade Level

Grades 8-10