Answered

Simplify the following expressions:


  1. $m^6 \cdot m^6$

Work / Add Exponents:

Answer: ____


  1. $p \cdot p^4$ (Hint: What exponent is invisibly on the first $p$?)

Work / Add Exponents:

Answer: ____


  1. $2x^3 \cdot 4x^2$ (Hint: Multiply the big numbers, add the exponents!)

Multiply coefficients (2 · 4): ____

Add exponents (3 + 2): ____

Answer: ____



Part 2: Power to a Power $\left(\left(x^a\right)^b\right)$

Strategy: When raising a power to another power, multiply the exponents.

Verified Answer :

  • To multiply expressions with the same base, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
  • For \(m^6 \cdot m^6\), the result is \(m^{6+6} = \boxed{m^{12}}\).
  • For \(p \cdot p^4\), treat \(p\) as \(p^1\), resulting in \(p^{1+4} = \boxed{p^5}\).
  • For \(2x^3 \cdot 4x^2\), multiply the coefficients \(2 \cdot 4 = 8\) and add the exponents \(x^{3+2} = x^5\) to get \(\boxed{8x^5}\).

Explanation

  1. Understanding the Product Rule of Exponents

When multiplying algebraic expressions with the same base, we use the product rule of exponents. This rule states that for any base \(a\) and exponents \(m\) and \(n\), the product is given by:
\[a^m \cdot a^n = a^{m+n}\]
This rule tells us that we keep the same base and add the exponents together. We will apply this principle to solve each of the given expressions.

  1. Calculating m^6 \cdot m^6

For the expression \(m^6 \cdot m^6\), both terms share the same base \(m\). Using the product rule, we add the exponents:
\[m^6 \cdot m^6 = m^{6+6} = m^{12}\]
Thus, the result for the first expression is \(m^{12}\).

  1. Calculating p \cdot p^4

For the expression \(p \cdot p^4\), we must remember that a variable written without an explicit exponent has an implicit exponent of 1. So, \(p = p^1\). Applying the product rule:
\[p^1 \cdot p^4 = p^{1+4} = p^5\]
Thus, the result for the second expression is \(p^5\).

  1. Calculating 2x^3 \cdot 4x^2

For the expression \(2x^3 \cdot 4x^2\), we multiply the coefficients (the numerical values) and then apply the product rule to the variable part with the base \(x\).
First, multiply the coefficients: \(2 \cdot 4 = 8\).
Next, add the exponents for the variable \(x\): \(3 + 2 = 5\).
Combining these parts, we get:
\[(2 \cdot 4) \cdot (x^3 \cdot x^2) = 8 \cdot x^{3+2} = 8x^5\]
Thus, the result for the third expression is \(8x^5\).

Examples

In science and engineering, exponents are essential for dealing with very large or small numbers. For instance, if you are calculating the total area of a surface represented by dimensions that follow power laws, such as \(\text{Area} = \text{length} \times \text{width}\), where length is \(2x^3\) meters and width is \(4x^2\) meters, the area is \(2x^3 \cdot 4x^2 = 8x^5\) square meters. Understanding how to combine these terms simplifies complex modeling and calculations in physics, biology, and finance.

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