Question

practice problems
apply the product rule

1. write each expression with a single exponent.

a. $10^3 \cdot 10^9$
b. $10 \cdot 10^4$
c. $-10^{10} \cdot 10^7$
d. $(-10)^3 \cdot (-10)^3$
e. $10^5 \cdot 10^{12}$
f. $10^6 \cdot 10^6 \cdot 10^6$

Explanation:

Part a

Step1: Recall product rule for exponents

The product rule for exponents states that \(a^m \cdot a^n=a^{m + n}\) when \(a\) is a non - zero real number and \(m,n\) are integers. For the expression \(10^{3}\cdot10^{9}\), here \(a = 10\), \(m=3\) and \(n = 9\).

Step2: Apply the product rule

Using the formula \(a^m\cdot a^n=a^{m + n}\), we have \(10^{3}\cdot10^{9}=10^{3 + 9}\)

Step3: Simplify the exponent

\(3+9 = 12\), so \(10^{3}\cdot10^{9}=10^{12}\)

Step1: Rewrite \(10\) as a power of \(10\)

We know that \(10=10^{1}\). So the expression \(10\cdot10^{4}\) can be written as \(10^{1}\cdot10^{4}\)

Step2: Apply the product rule for exponents

Using the rule \(a^m\cdot a^n=a^{m + n}\) with \(a = 10\), \(m = 1\) and \(n=4\), we get \(10^{1}\cdot10^{4}=10^{1 + 4}\)

Step3: Simplify the exponent

\(1+4=5\), so \(10\cdot10^{4}=10^{5}\)

Step1: Apply the product rule for exponents

For the expression \(- 10^{10}\cdot10^{7}\), using the rule \(a^m\cdot a^n=a^{m + n}\) (here \(a = 10\), \(m = 10\) and \(n = 7\)), we have \(-10^{10}\cdot10^{7}=-10^{10 + 7}\)

Step2: Simplify the exponent

\(10 + 7=17\), so \(-10^{10}\cdot10^{7}=-10^{17}\)

Answer:

\(10^{12}\)

Part b