Simplify each exponential expression in Exercises 23–64. $x^{-5}\cdot x^{10}$

Verified step by step guidance

1. Recall the property of exponents that states when multiplying two expressions with the same base, you add their exponents: \(a^m \cdot a^n = a^{m+n}\).
2. Identify the base and the exponents in the expression \(x^{-5} \cdot x^{10}\). Here, the base is \(x\), and the exponents are \(-5\) and \(10\) respectively.
3. Apply the exponent addition rule by adding the exponents: \(-5 + 10\).
4. Write the expression with the new exponent: \(x^{-5 + 10} = x^{5}\).
5. The simplified form of the expression is \(x^{5}\).

Key Concepts

Laws of Exponents

The laws of exponents provide rules for simplifying expressions involving powers. For multiplication of like bases, add the exponents (e.g., x^a * x^b = x^(a+b)). This rule is essential for combining terms like x^(-5) and x^(10).

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent (x^(-n) = 1/x^n). Understanding this helps in simplifying expressions where exponents are negative, converting them into fractions if needed.

Simplification of Exponential Expressions

Simplification involves applying exponent rules to rewrite expressions in a simpler or more standard form. This includes combining like terms, reducing powers, and expressing answers without negative exponents when possible.