Question

follow the instructions below. write ( p^3 cdot p^4 ) without exponents. ( p^3 cdot p^4 = square ) fill in the blank. ( p^3 cdot p^4 = p^{square} )

Explanation:

First Part (Write \( p^3 \cdot p^4 \) without exponents)

Step1: Recall the meaning of exponents

\( p^3 \) means \( p \cdot p \cdot p \) (3 factors of \( p \)) and \( p^4 \) means \( p \cdot p \cdot p \cdot p \) (4 factors of \( p \)).

Step2: Multiply the two expressions

When we multiply \( p^3 \) and \( p^4 \), we are multiplying the factors together: \( (p \cdot p \cdot p) \cdot (p \cdot p \cdot p \cdot p) \).

Step3: Count the number of \( p \) factors

Counting the number of \( p \)s, we have \( 3 + 4 = 7 \) factors of \( p \), so \( p^3 \cdot p^4 = p \cdot p \cdot p \cdot p \cdot p \cdot p \cdot p \).

Second Part (Fill in the blank \( p^3 \cdot p^4 = p^{\square} \))

Step1: Recall the exponent rule for multiplication

The rule for multiplying exponential expressions with the same base is \( a^m \cdot a^n = a^{m + n} \), where \( a \) is the base and \( m, n \) are exponents.

Step2: Apply the rule to \( p^3 \cdot p^4 \)

Here, the base \( a = p \), \( m = 3 \), and \( n = 4 \). So we add the exponents: \( 3 + 4 = 7 \). Thus, \( p^3 \cdot p^4 = p^7 \).

Answer:

s:

• For the first blank (without exponents): \( p \cdot p \cdot p \cdot p \cdot p \cdot p \cdot p \)
• For the second blank (with exponent): \( 7 \)