- Apply the product rule for exponents: \(x^a \cdot x^b = x^{a+b}\).
- Calculate the new exponent: \(3 + (-3) = 0\).
- Recognize the identity \(x^0 = 1\) for non-zero \(x\).
- The simplified expression is \(x^0 = 1\).
- Final Answer: \(\boxed{0, 1}\)
Explanation
- Analyzing the expression and the product rule
The expression we are analyzing is \(x^3 \cdot x^{-3}\). We want to rewrite this in the form \(x^n\) and find its numerical value. According to the product rule for exponents, when we multiply two powers with the same base, we add their exponents: \(x^a \cdot x^b = x^{a+b}\). In our case, the base is \(x\), and the exponents are \(3\) and \(-3\).
- Simplifying the exponent
To simplify the expression, we add the exponents as per the product rule: \(3 + (-3) = 0\). Therefore, the expression becomes \(x^3 \cdot x^{-3} = x^0\). The exponent is \(0\).
- Evaluating the expression
Any non-zero real number raised to the power of \(0\) is defined as \(1\). Thus, \(x^0 = 1\). The final value of the expression is \(1\).
Examples
Understanding the laws of exponents is fundamental for scientific notation and advanced algebra. For example, if you are working with very large or very small measurements in physics, such as calculating the ratio of two forces acting in opposite directions at the same scale, you often end up with an expression like \(10^3 \cdot 10^{-3}\). Simplifying this to \(1\) helps significantly in scaling and comparing data values efficiently.