Problem 27 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. \(\left[(-9)^{4}\right]^{3}=(-9)^{?}\)

Step-by-Step Solution

Answer

The final expression is \( (-9)^{12}.

Step 1: Identifying the numbers

First, identify the numbers that you will be working with. Here, the base is -9, the power of the base is 4, and the power of the whole expression is 3.

Step 2: Applying the power of powers rule

Next, apply the power of powers rule, which states that \( (a^{m})^{n} = a^{mn} \). So, in this case, \( \left[(-9)^{4}\right]^{3}=(-9)^{(4*3)}.

Step 3: Simplifying the exponent

Now, simplify the exponent to get the final answer. In this case, \( (-9)^{(4*3)}=(-9)^{12}.

Key Concepts

Exponents

When we talk about exponents in mathematics, we're referring to the shorthand notation for expressing repeated multiplication of the same number. For example, if we have the number 5 and want to multiply it by itself three times, we write this as 5^3, where '5' is the base and '3' is the exponent. Understanding exponents is crucial because they are not only ubiquitous in different areas of math like algebra and calculus, but also in various scientific fields.

Exponents have their own set of rules for operations such as multiplication, division, and raising powers to powers. When multiplied together, bases that are the same can have their exponents added, while division allows us to subtract exponents. It's these rules that help us simplify complex algebraic expressions and solve equations efficiently.

Simplifying Exponents

Simplifying exponents, also known as simplifying powers, is a process that makes an expression with exponents easier to work with. For example, the power of powers rule comes in handy when dealing with expressions like (a^m)^n. Instead of multiplying the base 'a' by itself 'm' times and then taking that result and multiplying it 'n' times, we can simply multiply the exponents together to get a^(m*n).

This simplification reduces the complexity of calculations and is particularly useful when dealing with large numbers or variables. Always remember that in order to apply this rule, the base must remain the same. Simplifying exponents allows us to condense and solve expressions that would otherwise be time-consuming to compute.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, operators, and variables. Variables are symbols like x, y, or z that represent unknown values. Algebraic expressions can be as simple as x + 5 or as complex as (3x^2 - 2xy + y)/(7x - y^2). The beauty of algebra is in the ability to manipulate these expressions using established rules to simplify or solve them.

For instance, knowing how to work with exponents will enable us to condense and simplify expressions, making them more manageable. Solving algebraic expressions often involves simplifying the parts of the expression, isolating the variable, and finally solving for that variable. As we do this, we apply various algebra rules, such as the distributive property, combining like terms, and the power of powers rule we've discussed earlier.