Math Problem Statement
The expression below can be simplified by the laws of indices to one of the 5 expression options or "None" (which is also an option). Pick the correct answer.
(c3/2h-1b2k1)2(k-3/2b1)1(h1b1c3/2)-2
Select one:
b3c-1h-4k3/2
b3h-4k1/2
b4h-4k3/2
b3h-4k5/2
b3h-3k3/2
None
Solution
To simplify the given expression, let us carefully apply the laws of indices step by step. Here's the expression:
Step 1: Expand each term using the laws of indices
-
Expand :
- Each exponent is multiplied by :
-
Expand :
- Raising to the power of leaves the expression unchanged:
-
Expand :
- Each exponent is multiplied by :
Step 2: Combine the terms in the numerator
The numerator is:
- Combine the powers of , , , and using the product rule ():
- :
- :
- :
- :
Thus, the numerator simplifies to:
Step 3: Simplify the denominator
The denominator is:
Step 4: Divide the numerator by the denominator
Using the quotient rule ():
- : (cancels out completely)
- :
- :
- : (no in the denominator, so it remains )
The final simplified expression is:
Step 5: Match the result with the options
This matches:
Do you want further clarifications or detailed explanation?
Here are 5 related questions for practice:
- How do you handle negative exponents when simplifying fractions?
- Why does when simplifying expressions?
- How can you simplify expressions involving multiple variables?
- What are the key differences between product and quotient rules for exponents?
- How do fractional exponents (e.g., ) relate to roots?
Tip: Always combine terms systematically, keeping track of the variables one by one to avoid mistakes.
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Math Problem Analysis
Mathematical Concepts
Laws of Indices
Exponents
Algebraic Simplification
Formulas
Product Rule: a^m * a^n = a^(m+n)
Quotient Rule: a^m / a^n = a^(m-n)
Power Rule: (a^m)^n = a^(m*n)
Theorems
Exponential Laws for Algebraic Expressions
Suitable Grade Level
Grades 9-11
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