Question

Multiply and simplify. $$(2.25 a)\left(1.55 a^{m}\right)\left(2.36 a^{n}\right)$$

Short answer

The simplified result is 8.214a^{1+m+n}.

Step-by-step solution

Multiply the coefficients

First, multiply the numerical coefficients 2.25, 1.55, and 2.36 together.

Multiply the variables

Then, multiply the variables together. Use the property of exponents that states when you multiply like bases you add the exponents. Here the base is 'a', the exponents are 1, m, and n. The sum of the exponents is thus 1 + m + n.

Combine the results

Combine the product of the coefficients from Step 1 with the product of the variables from Step 2 to get the simplified expression.

Key concepts

Multiplying Coefficients

When you come across an algebraic problem that requires you to multiply coefficients, think of it as a regular multiplication task, the same kind you learned in basic arithmetic. The beauty of algebra does not change the core principles of multiplication. For instance, take the case of multiplying three numbers, 2.25, 1.55, and 2.36. You multiply these numbers exactly as you'd multiply any other numbers.

Here's a simplified process you might follow:

• Lay out the numbers: 2.25, 1.55, 2.36.
• Use a calculator or perform long multiplication if you're doing it by hand.
• First, you might round the numbers to make initial calculations easier, but ensure you return to the exact numbers for the final answer.
• The product of these coefficients will give you a single numerical coefficient that you will then use alongside the variable part of the expression.

This process is a fundamental part of simplifying algebraic expressions and should be mastered early on because it is a step that cannot be bypassed and is integral in reaching the correct solution.

Properties of Exponents

Understanding the properties of exponents is crucial when dealing with algebraic expressions that involve powers. One key property to remember is that when you are multiplying terms with the same base, you should add the exponents. In our example, you're given the base 'a' with different exponents.

Here's how you apply this property:

• Identify like bases in the terms you are multiplying. In our case, each group of terms contains the base 'a'.
• Add the exponents together keeping the base same. For example, if you have \(a\) and \(a^m\), when you multiply them, you add the 1 (the implicit exponent for \(a\)) and \(m\) to get \(a^{1+m}\).
• If you had another term, say \(a^n\), you would further add \(n\) to get \(a^{1+m+n}\).

The simplicity of this rule allows it to be universally applicable in all algebraic contexts where multiplication of like bases is involved. It is important not to confuse this property with others, such as the power of a power or multiplying bases, which have their own specific rules.

Combining Like Terms

Combining like terms is one of the final steps in the process of simplifying algebraic expressions. This step is vital in achieving the simplest form of an expression. 'Like terms' are terms that have exactly the same variable parts, meaning the same variables raised to the same powers.

Here's what you should remember when combining like terms:

• Look for and group together like terms that have the same variable factors.
• Add or subtract the coefficients of these terms as indicated (though in our example, we're only dealing with multiplication).
• Ensure that the variable part remains unchanged after combining the coefficients.

In the exercise provided, once you have multiplied the coefficients and applied the properties of exponents, you will end up with a single term. This streamlines the expression and shows the power of combining like terms in simplifying algebraic expressions to their most accessible form.