Exponential Functions - Definitions

Question 1: True Statements about Exponent Laws (from file image_c68f8a.png)

The Question: Which of the following options is(are) TRUE? (Multiple Select Question)

Detailed Solution:

Let’s evaluate each statement using the laws of exponents.

• a^s \times a^t = a^{s+t}: This is the correct Product Rule. The conditions given, for $s, t \in \mathbb{R}$ and $a > 0$, are the standard domain for this rule to apply universally. This statement is TRUE.

• (a^s)^t = a^{s+t}: This is FALSE. This incorrectly states the Power of a Power Rule. The exponents should be multiplied, not added. The correct rule is $(a^s)^t = a^{s \times t}$.

• (ab)^s = a^s \times b^s: This is the correct Power of a Product Rule. This statement is TRUE.

• a^s \times b^s = (a+b)^s: This is FALSE. This is a common algebraic mistake. You can combine the bases when they are multiplied ($(a \times b)^s$), but not when they are added.

• a^s \times a^t = a^{s+t}: This is a repeat of the first option and is TRUE.

Final Answer: The true statements are:

• $(ab)^s = a^s \times b^s$, for $s \in \mathbb{R}$ and $a,b > 0$
• $a^s \times a^t = a^{s+t}$, for $s, t \in \mathbb{R}$ and $a > 0$.

Question 2: Correct Equations (from file image_c68f8a.png)

The Question: Which of the following equations is(are) CORRECT? (Multiple Select Question)

Detailed Solution:

Let’s apply the laws of exponents to check each equation.

• 2^3 \times 2^4 = 2^12: FALSE. According to the Product Rule, when multiplying powers with the same base, we must add the exponents: $2^3 \times 2^4 = 2^{3+4} = 2^7$.

• 2^3 \times 2^4 = 2^7: TRUE. This correctly applies the Product Rule.

• 3^5 \times 9^3 = 3^11: TRUE. To solve this, we must first express both terms with the same base. Since $9 = 3^2$, we can rewrite the equation:

  • $3^5 \times (3^2)^3$
  • Apply the Power of a Power Rule: $3^5 \times 3^{2 \times 3} = 3^5 \times 3^6$
  • Apply the Product Rule: $3^{5+6} = 3^{11}$.

• 7^3 \times 5^3 = 35^9: FALSE. When multiplying powers with the same exponent but different bases, we multiply the bases and keep the exponent: $7^3 \times 5^3 = (7 \times 5)^3 = 35^3$.

Final Answer: The correct equations are:

• $2^3 \times 2^4 = 2^7$
• $3^5 \times 9^3 = 3^{11}$

Question 3: Simplifying a Complex Expression (from file image_c68f0f.png)

The Question: Simplify the expression $(2^2 \times 3^3)^5 \times 4^7 \times 5^3 \times (8 \times 25^3)^2$.

Detailed Solution:

The strategy is to break down every number into its prime factors (2, 3, 5, etc.) and then use the exponent rules to combine them.

1. Break down each part of the expression:

   • Part 1: $(2^2 \times 3^3)^5 = (2^2)^5 \times (3^3)^5 = 2^{10} \times 3^{15}$
   • Part 2: $4^7 = (2^2)^7 = 2^{14}$
   • Part 3: $5^3$ (already a prime base)
   • Part 4: $(8 \times 25^3)^2 = (2^3 \times (5^2)^3)^2 = (2^3 \times 5^6)^2 = (2^3)^2 \times (5^6)^2 = 2^6 \times 5^{12}$

2. Combine all the simplified parts:

   • $(2^{10} \times 3^{15}) \times (2^{14}) \times (5^3) \times (2^6 \times 5^{12})$

3. Group the terms by their base:

   • $(2^{10} \times 2^{14} \times 2^6) \times (3^{15}) \times (5^3 \times 5^{12})$

4. Add the exponents for each base (Product Rule):

   • $2^{10+14+6} \times 3^{15} \times 5^{3+12} = 2^{30} \times 3^{15} \times 5^{15}$

5. Look for a common exponent to simplify further:

   • Notice that the exponent 15 is common. We can rewrite $2^{30}$ as $(2^2)^{15} = 4^{15}$.
   • The expression is now: $4^{15} \times 3^{15} \times 5^{15}$

6. Combine the bases that have the same exponent:

   • $(4 \times 3 \times 5)^{15} = (12 \times 5)^{15} = 60^{15}$

Final Answer: The simplified expression is $60^{15}$.

Question 4: Identifying Incorrect Statements (from file image_c68f0f.png)

The Question: Which of the following statements is(are) INCORRECT? (Multiple Select Question)

Detailed Solution:

We are looking for the statements that are false.

• “Every exponential function is a one-to-one function.”

  • An exponential function, $f(x)=b^x$ (where $b>0, b \neq 1$), is either always increasing (if $b>1$) or always decreasing (if $0<b<1$). In either case, it passes the Horizontal Line Test. This statement is TRUE.

• "$0^0$ is undefined."

  • In most algebraic and calculus contexts, $0^0$ is considered an “indeterminate form” and is left undefined because its value depends on the context from which it arises. This statement is TRUE.

• "$a^0 = 1$, for all $a \in \mathbb{R}$."

  • This statement is FALSE. The zero exponent rule applies to all non-zero numbers, but it does not apply to $a=0$. As stated above, $0^0$ is undefined.

• "a is the exponent in the algebraic expression $a^r$."

  • This statement is FALSE. In the expression $a^r$, ‘$a$’ is called the base and ‘$r$’ is the exponent (or power).

Final Answer: The incorrect statements are:

• $a^0 = 1$, for all $a \in \mathbb{R}$.
• $a$ is the exponent in the algebraic expression $a^r$.

Question 5: Identifying a Non-Exponential Function (from file image_c68c2d.png)

The Question: Which of the following options is not an exponential function?

• $f(x) = 3^{x/2}$
• $f(x) = x^{9/4}$
• $f(x) = \frac{15}{8^x}$
• $f(x) = 20 \times 6^x$

Core Concept: The defining characteristic of an exponential function is that the variable (like $x$) is in the exponent. If the variable is the base, it’s called a power function.

Detailed Solution:

Let’s examine each function:

• $f(x) = 3^{x/2}$: The variable $x$ is in the exponent. This is an exponential function.
• $f(x) = x^{9/4}$: The variable $x$ is the base. This is a power function, not an exponential function.
• $f(x) = \frac{15}{8^x}$: We can rewrite this as $f(x) = 15 \times 8^{-x} = 15 \times (\frac{1}{8})^x$. The variable $x$ is in the exponent. This is an exponential function.
• $f(x) = 20 \times 6^x$: The variable $x$ is in the exponent. This is an exponential function.

Final Answer: The function that is not an exponential function is $f(x) = x^{9/4}$.