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Product Rule		$x ^{ a } \cdot x ^{ b }$ADD the exponents$x^3 \cdot x^4=x^{3+4}=x^7$

(Power × Power)

Power Rule (Power to a Power)		$\left(x^a\right)^b$MULTIPLY the exponents$\left(x^3\right)^4=x^{3 \cdot 4}=x^{12}$






Part 1: Product of Powers $\left(x^a \cdot x^b\right)$

Strategy: When multiplying terms with the same base, add the exponents.

Helpful Tip: Expand the factors if you get stuck!



Guided Example: $y^2 \cdot y^3$

  • Expanded Form: $(y \cdot y) \cdot (y \cdot y \cdot y)$
  • Count the total y's: 5
  • Final Answer: $y^5$


  1. $a^4 \cdot a^3$

Write in expanded form: \_\_\_\_

Work / Add Exponents:

Answer: \_\_\_\_


  1. $x^5 \cdot x^2$

Work / Add Exponents:

Answer: \_\_\_\_

Verified Answer :

  • The product rule for exponents \(x^a \cdot x^b = x^{a+b}\) allows us to simplify expressions with the same base by adding their exponents.
  • For \(a^4 \cdot a^3\), the expanded form is \((a \cdot a \cdot a \cdot a) \cdot (a \cdot a \cdot a) = a^7\), and the exponent sum is \(4+3=7\).
  • For \(x^5 \cdot x^2\), the exponent sum is \(5+2=7\), resulting in \(x^7\).
  • Final answers are \(a^7\) and \(x^7\), respectively, denoted as \(\boxed{{a^7, x^7}}\).

Explanation

  1. Simplifying the Expression \(a^4 \cdot a^3\)

To simplify the expression \(a^4 \cdot a^3\), we observe that we are multiplying two powers with the same base \(a\). The product rule for exponents states that for any non-zero base \(x\) and integers \(m\) and \(n\), \(x^m \cdot x^n = x^{m+n}\).

First, we write \(a^4\) and \(a^3\) in expanded form:
\(a^4 = a \cdot a \cdot a \cdot a\)
\(a^3 = a \cdot a \cdot a\)

When we multiply these two, we combine all the factors:
\(a^4 \cdot a^3 = (a \cdot a \cdot a \cdot a) \cdot (a \cdot a \cdot a) = a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a\)

Counting the total number of \(a\)'s, we find there are 7. Using the product rule, we add the exponents directly: \(4 + 3 = 7\). Thus, the simplified form is \(a^7\).

  1. Simplifying the Expression \(x^5 \cdot x^2\)

To simplify the expression \(x^5 \cdot x^2\), we again apply the product rule for exponents: \(x^m \cdot x^n = x^{m+n}\).

In this case, the base is \(x\) and the exponents are 5 and 2. Summing these exponents gives us:
\(5 + 2 = 7\)

Therefore, by the product rule for exponents, we have:
\(x^5 \cdot x^2 = x^{5+2} = x^7\)

The final simplified expression is \(x^7\).

Examples

Understanding exponent rules is essential in fields like computer science, chemistry, and physics. For example, when calculating how memory storage grows in a system, or how radioactive decay affects the concentration of a substance, you often deal with powers of the same base. By knowing that \(10^2 \cdot 10^3 = 10^5\), you can quickly simplify complex calculations, saving time and preventing errors in real-world applications where data scales exponentially.

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