Problem 122 - Algebra A Combined Function

Question

Simplify. $$ \left(2 a^{3}\right)^{3} a^{-3}+a^{11} a^{-5} $$

Step-by-Step Solution

Answer

The simplified expression is \( 9 a^6 \).

Step 1: Understand the Expression

The given expression is \( \left(2 a^{3}\right)^{3} a^{-3} + a^{11} a^{-5} \). It contains two parts separated by a plus sign. We will simplify each part separately.

Step 2: Use the Power of a Power Rule

For the term \( \left(2 a^{3}\right)^{3} \), apply the power of a power rule: \((x^m)^n = x^{m \cdot n}\). Simplifying, \( \left(2 a^{3}\right)^{3} = 2^3 a^{3 \cdot 3} = 8 a^9 \).

Step 3: Use the Product of Powers Rule

Now, consider \( 8 a^9 a^{-3}\). Use the product of powers rule \( a^m a^n = a^{m+n} \). Therefore, combining the powers of \(a\), we have \( 8 a^{9-3} = 8 a^6 \).

Step 4: Simplify the Second Part

Simplify \( a^{11} a^{-5} \) using the product of powers rule \( a^m a^n = a^{m+n} \), so \( a^{11} a^{-5} = a^{11-5} = a^6 \).

Step 5: Combine Like Terms

Combine the results from both parts: \( 8 a^6 + a^6 \). These are like terms. Add their coefficients: \( (8+1) a^6 = 9 a^6 \).

Step 6: Final Simplified Expression

The simplified form of the original expression is \( 9 a^6 \).

Key Concepts

Power of a Power Rule

When dealing with exponents, we often encounter situations where exponents are raised to another power. This is where the "power of a power rule" becomes quite handy. This rule states that when you have a power raised to another power, you multiply the exponents. For instance, \[(x^m)^n = x^{m \cdot n}\]This means you should multiply the exponent of the base by the exponent of the power outside the parentheses. In our exercise, the term \((2a^3)^3\) applies this rule. After applying it, we simplify to get:

• \(2^3 = 8\)
• \(a^{3 \times 3} = a^9\)

So, \((2a^3)^3 = 8a^9\). This rule simplifies calculations involving exponents drastically.

Product of Powers Rule

Another important concept in working with exponents is the "product of powers rule." This rule is used when you multiply two expressions with the same base. According to this rule:\[a^m \cdot a^n = a^{m+n}\]You simply keep the base the same and add the exponents together, making this process straightforward. In the initial step of simplifying the exercise:When dealing with the term \(8 a^9 \cdot a^{-3}\)Using the product of powers rule, you add 9 and -3. This simplifies to: \[8a^6\]This makes solving expressions with exponents swift and less confusing.

Like Terms

"Like terms" are crucial in simplifying algebraic expressions. They are terms that have the same variables raised to the same powers, allowing them to be combined.For example, in our exercise:You get \(8a^6\) from one part and \(a^6\) from another. Both have the exact same base, \(a^6\), which means they are like terms.Consequently, you can add them together by merely adding their coefficients:

• \(8 + 1 = 9\)

Thus, they combine seamlessly into \(9a^6\). Understanding like terms greatly simplifies combining different parts of an expression.

Exponents and Powers

Exponents and powers are fundamental aspects of algebra. They express how many times a number, the base, is multiplied by itself.For example:

• In \(a^3\), \(a\) is the base and 3 is the exponent. It means \(a\) is multiplied by itself three times: \(a \cdot a \cdot a\).

Understanding the nature of exponents is crucial, as they appear frequently in mathematics. They help describe large numbers succinctly and solve complex problems efficiently.In the original problem, handling the terms involving exponents was pivotal in breaking down and then combining the expression into its simplest form. This thorough comprehension of exponents and powers makes manipulating and simplifying expressions straightforward.