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In algebra, when you multiply two numbers with the same base and different exponents, you add the exponents together. This is known as the product of powers property. For example, if we have: \(a^m \times a^n\), we can combine these terms into a single term by adding the exponents: \(a^{m+n}\).

In the given exercise: \(m^{-3} \times m^4\), the base 'm' is the same for both terms. Using the product of powers property, we add the exponents: \(-3 + 4\). This simplifies to: \(m^{1} = m\).

This property makes it easier to simplify expressions involving the same base with different exponents. It's particularly useful in algebra to keep expressions neat and manageable.

Exponents are a fundamental part of algebra and are used to express repeated multiplication of a number. For example, \(b^7\) means \(b\) is multiplied by itself 7 times: \(b \times b \times b \times b \times b \times b \times b\).

There are a few key rules to remember when working with exponents:

• The product of powers property: \(a^m \times a^n = a^{m+n}\)
• The power of a power property: \((a^m)^n = a^{m \times n}\)
• The power of a product property: \((ab)^m = a^m \times b^m\)

Understanding these rules will help you simplify expressions and solve algebraic equations efficiently. In our exercise, we utilized the product of powers property to simplify \(m^{-3} \times m^4 = m\), and combined it with the independent term \(b^7\).

Simplifying algebraic expressions involves reducing them to their most basic form without changing their value. This usually means combining like terms and applying algebraic rules such as the product of powers property.

In our original exercise, we started with the expression: \(m^{-3} \times m^4 \times b^7\).

We first applied the product of powers property to the terms with the same base (\(m\)) and combined the exponents: \(m^{-3+4} = m\).

The resultant expression \(m \times b^7\) is already in its simplest form since there are no additional like terms to combine.

Simplifying expressions makes them easier to work with, especially in more complex algebraic operations. It helps in solving equations, factoring, and performing other algebraic manipulations.