Problem 133 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

Step-by-Step Solution

Answer

The given statement \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49\) is FALSE. The correct statement should be \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=7\)

Step 1: Identify the exponential rule

Identify that \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}\) utilizes the rule that when you multiply like bases, you add the exponents. In this case, the base is 7 and the exponents are both \(\frac{1}{2}\).

Step 2: Apply the exponential rule

By the multiplication property of exponents, \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=7^{\frac{1}{2}+\frac{1}{2}}=7^1\)

Step 3: Final calculation and verification

Further determine that \(7^1=7\). The statement \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49\) is false. To make it true, it should be corrected to \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=7\)

Key Concepts

Exponents

The concept of exponents is a foundational part of algebra and mathematics as a whole. An exponent indicates how many times a number, called the base, is multiplied by itself. For instance, in the expression \(7^3\), 7 is the base, and 3 is the exponent, meaning \(7\) is multiplied by itself three times, which is \(7 \times 7 \times 7\).When dealing with exponents, it’s important to remember some key rules:

• \(a^0 = 1\), where \(a\) is any non-zero number.
• \(a^1 = a\), since any number to the power of 1 is itself.
• \(a^n \cdot a^m = a^{n+m}\), which shows that exponents are added when multiplying like bases.

These rules help simplify complex expressions and should be always considered when working through algebraic problems.

Multiplication Property

The multiplication property of exponents is an essential rule that facilitates the simplification of expressions with like bases. According to this property, when you multiply two exponential expressions that have the same base, you add the exponents.Let’s say you have \(x^a \cdot x^b\). According to the multiplication property, the result will be \(x^{a+b}\).For example:

• \(3^2 \cdot 3^3 = 3^{2+3} = 3^5\)
• \(y^7 \cdot y^1 = y^{7+1} = y^8\)

By applying this property, we can reduce many algebraic expressions to simpler, more manageable forms.

Like Bases

In algebra, the term "like bases" refers to expressions that have the same base in an exponential term, such as \(a^n\) \(\text{and}\) \(a^m\), where \(a\) is the base. Understanding and identifying like bases is crucial because it allows for the application of exponential rules, like the multiplication property.Why are like bases important?

• They simplify calculations: You can only apply the rule of adding exponents if the bases are alike.
• They help in factorization: Recognizing like bases can assist in breaking down more complex algebraic expressions.

Consider the expression \(2^3 \cdot 2^5\), as the bases are both \(2\), the expression simplifies neatly to \(2^{3+5} = 2^8\). This simplification would not be possible if the bases were different.

False Statements in Algebra

In algebra, identifying and correcting false statements is a vital skill. False statements often occur due to misunderstanding or misapplying mathematical rules, such as those regarding exponents.In the provided exercise, the initial statement \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 49\) is false because the rule of multiplication of exponents was not applied correctly.Key steps to verify statements:

• Re-evaluate the use of rules: Ensure the correct application of exponential rules.
• Check calculations: Verify each step by recalculating if necessary.
• Look at the logical outcome: Does the result make sense?

By following these steps, you can correct false statements, like changing \(7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}} = 49\) to \(7^{1} = 7\), ensuring accuracy in algebraic problem-solving.