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Understanding how to multiply like bases is key in simplifying expressions with exponents. When you are given an expression like \( a^3 \cdot a^3 \), it means you are multiplying two powers that have the same base, which is \(a\) in this case. The rule for multiplying like bases states that you simply add the exponents while keeping the base the same. In our expression, you add the exponents:

• \( 3 \) from the first \( a^3 \)
• \( 3 \) from the second \( a^3 \)

Thus, the simplified expression is \( a^{3+3} = a^6 \).
This process works regardless of what the base is, as long as they are the same. So, for any like bases \( b^m \cdot b^n = b^{m+n} \), simply add the exponents together.

Mastering the power of a power property is important when dealing with exponential expressions that are nested. Consider the expression \((a^3)^3\). This type of expression involves raising an exponent to an additional power.The power of a power property states that you multiply the exponents. Here is the step-by-step approach:

• Start with the inner exponent, which is \(a^3\).
• Then, since we are raising \(a^3\) to the third power, we multiply the exponents: \( 3 \times 3 \).

This results in \( a^{3 imes 3} = a^9 \).
By following this method, you ensure the expression is simplified correctly. For an expression like \((b^5)^4\), simply calculate \(5 \times 4\) to get \(b^{20}\). Always remember to multiply, not add, the exponents for nested powers.

Factoring common terms is a great way to simplify expressions when addition is involved. Consider the expression \( a^3 + a^3 \). Here, instead of multiplication, you have addition, which changes the approach.You can factor out the common term \( a^3 \) from the expression. Think of each \( a^3 \) as being multiplied by 1:

• First term: \(1a^3\)
• Second term: \(1a^3\)

Combine them using the distributive property:
\( a^3 + a^3 = 1a^3 + 1a^3 \) becomes \((1+1) \cdot a^3 = 2a^3\).This technique uses the idea of factoring out what's common or similar in each term, effectively simplifying the expression to \(2a^3\). Whenever you notice identical terms being added, look for opportunities to factor them out to streamline your expression.