Rewrite Each Expression as a Single Power: A Guide to Simplifying Exponents
Rewriting exponential expressions as a single power is a foundational skill in algebra that helps streamline complex calculations and solve equations more efficiently. By applying the laws of exponents, you can condense multiple terms with the same base into one simplified expression. This technique is essential for students progressing in mathematics and anyone looking to master algebraic manipulation.
Steps to Rewrite Expressions as a Single Power
Step 1: Identify the Base and Exponents
First, determine if the terms share the same base. To give you an idea, in the expression $ x^2 \cdot x^3 $, the base is $ x $, and the exponents are 2 and 3. If the bases differ, this method may not apply directly And it works..
Step 2: Apply the Appropriate Exponent Rule
Use one of the following rules based on the operation involved:
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Product of Powers: When multiplying terms with the same base, add the exponents.
$ a^m \cdot a^n = a^{m+n} $
Example: $ 2^4 \cdot 2^5 = 2^{4+5} = 2^9 $ -
Quotient of Powers: When dividing terms with the same base, subtract the exponents.
$ \frac{a^m}{a^n} = a^{m-n} $ (where $ a \neq 0 $)
Example: $ \frac{3^7}{3^2} = 3^{7-2} = 3^5 $ -
Power of a Power: When raising a power to another exponent, multiply the exponents.
$ (a^m)^n = a^{m \cdot n} $
Example: $ (5^2)^3 = 5^{2 \cdot 3} = 5^6 $ -
Zero Exponent Rule: Any non-zero base raised to the power of 0 equals 1.
$ a^0 = 1 $ (where $ a \neq 0 $)
Example: $ 7^0 = 1 $ -
Negative Exponent Rule: A negative exponent indicates the reciprocal of the base raised to the positive exponent.
$ a^{-n} = \frac{1}{a^n} $ (where $ a \neq 0 $)
Example: $ 4^{-2} = \frac{1}{4^2} = \frac{1}{16} $
Step 3: Combine and Simplify
After applying the rule, combine the results into a single term. If possible, reduce the final exponent to its simplest form And it works..
Scientific Explanation of the Laws
The laws of exponents are rooted in the definition of exponentiation as repeated multiplication. Because of that, for instance, $ x^3 \cdot x^2 $ means multiplying $ x $ three times and then two more times, totaling five $ x
