Which expression is equivalent to (10x)^-3

Equivalent Expression for $(10x)^{-3}$

Let's determine the equivalent expression for $(10x)^{-3}$.

Starting with the given expression:

We can use the properties of exponents to rewrite it:

Now, let's expand the denominator:

Thus, the expression becomes:

Therefore, the equivalence is:

Conclusion

The equivalent expression for $(10x)^{-3}$ is:

Importance of Exponent Rules

Exponent rules are fundamental principles in mathematics that govern how to perform operations involving exponents. These rules simplify complex expressions and are critical for solving various algebraic problems.

Basic Rules of Exponents

1. Product of Powers When multiplying two expressions with the same base, you add the exponents.

   $$a^m \times a^n = a^{m+n}$$

2. Quotient of Powers When dividing two expressions with the same base, you subtract the exponents.

   $$\frac{a^m}{a^n} = a^{m-n}$$

3. Power of a Power When raising an exponent to another exponent, you multiply the exponents.

   $$(a^m)^n = a^{mn}$$

4. Product of Powers with Different Bases When multiplying expressions with different bases but the same exponent, you can multiply the bases first and then raise to the power.

   $$a^n \times b^n = (a \times b)^n$$

5. Zero Exponent Any non-zero base raised to the zero power is equal to 1.

   $$a^0 = 1 \quad \text{for} \quad a \neq 0$$

6. Negative Exponent A negative exponent represents the reciprocal of the base raised to the opposite positive exponent.

   $$a^{-n} = \frac{1}{a^n}$$

Rational Exponents

Rational exponents extend the idea of exponents to fractions. A rational exponent represents both an exponent and a root.

Importance in Algebra

Exponent rules are essential in simplifying algebraic expressions, solving exponential equations, and understanding the growth patterns in various scientific fields such as physics, biology, and economics. By mastering these rules, you can efficiently handle complex calculations involving powers and roots.

Product of Powers

When you multiply numbers with the same base, you add their exponents. If $a$ is any non-zero number and $m$ and $n$ are integers, then:

Quotient of Powers

When you divide numbers with the same base, you subtract the exponent of the denominator from the exponent of the numerator:

Power of a Power

When raising an exponent to another exponent, you multiply the exponents:

Power of a Product

When raising a product to an exponent, you distribute the exponent to both bases:

Power of a Quotient

When raising a quotient to an exponent, you distribute the exponent to both the numerator and the denominator:

Zero Exponent

Any non-zero base raised to the zero power is equal to one:

Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent:

Fractional Exponent

A fractional exponent indicates both a root and a power. Specifically, if the exponent is $\frac{m}{n}$, it means raise the base to the power of $m$ and then take the $n$th root:

Understanding these properties of exponents can greatly help in simplifying expressions and solving equations that involve exponents.