Math Problem Statement

Simplify the expression completely: (3z^4 * 4z^9) / 8z^10

Solution

Let's simplify the given expression:

3z44z98z10\frac{3z^4 \cdot 4z^9}{8z^{10}}

Step-by-step:

  1. First, simplify the constants: 348=128=32\frac{3 \cdot 4}{8} = \frac{12}{8} = \frac{3}{2}

  2. For the powers of zz, use the property of exponents: zazb=za+bz^a \cdot z^b = z^{a + b} Applying this: z4z9=z4+9=z13z^4 \cdot z^9 = z^{4 + 9} = z^{13}

  3. Now, the expression is: 3z132z10\frac{3z^{13}}{2z^{10}}

  4. Use the quotient rule for exponents: zazb=zab\frac{z^a}{z^b} = z^{a - b} Applying this: z13z10=z1310=z3\frac{z^{13}}{z^{10}} = z^{13 - 10} = z^3

Final result:

32z3\frac{3}{2} z^3

This is the simplified expression.

Would you like further details or explanations?

Here are some related questions:

  1. How do we apply the quotient rule for exponents?
  2. What happens if the exponents are negative?
  3. How would the answer change if the exponents were fractional?
  4. Can you explain how to handle multiplication and division of terms with different bases?
  5. What are some common mistakes made when simplifying expressions with exponents?

Tip: Always simplify constants separately from variables to avoid confusion.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponents
Simplification

Formulas

Product of Powers Rule: z^a * z^b = z^{a+b}
Quotient of Powers Rule: z^a / z^b = z^{a-b}

Theorems

Exponent Laws
Simplifying Algebraic Expressions

Suitable Grade Level

Grades 9-12