Question

finding equivalent expressions

1. store a number for variable. (choose a number greater than 3)
2. number, ctrl, var, variable.
3. set the original equation equal to one of the choices.
4. is it true or false? check each choice till you find the one that is true!

1. a computer application generates a sequence of musical notes using the function (f(n) = 6(16)^n), where (n) is the number of the note in the sequence and (f(n)) is the note frequency in hertz. which function will generate the same note sequence as (f(n))?
2. (g(n) = 12(2)^{4n})
3. (h(n) = 6(2)^{4n})
4. (p(n) = 12(4)^{2n})
5. (k(n) = 6(8)^{2n})

1. the function (f(x) = 3x^2 + 12x + 11) can be written in vertex form as
2. (f(x) = (3x + 6)^2 - 25)
3. (f(x) = 3(x + 6)^2 - 25)
4. (f(x) = 3(x + 2)^2 - 1)
5. (f(x) = 3(x + 2)^2 + 7)

1. marios \\$15,000 car depreciates in value at a rate of 19% per year. the value, (v), after (t) years can be modeled by the function (v = 15,000(0.81)^t). which function is equivalent to the original function?
2. (v = 15,000(0.9)^{9t})
3. (v = 15,000(0.9)^{2t})
4. (v = 15,000(0.9)^{\frac{t}{9}})
5. (v = 15,000(0.9)^{\frac{t}{2}})

1. nora inherited a savings account that was started by her grandmother 25 years ago. this scenario is modeled by the function (a(t) = 5000(1.013)^{t+25}), where (a(t)) represents the value of the account, in dollars, (t) years after the inheritance. which function below is equivalent to (a(t))?
2. (a(t) = 5000(1.013)^t^{25})
3. (a(t) = 5000(1.013)^t + (1.013)^{25})
4. (a(t) = (5000)^t(1.013)^{25})
5. (a(t) = 5000(1.013)^t(1.013)^{25})

1. the number of bacteria grown in a lab can be modeled by (p(t) = 300 \cdot 2^{4t}), where (t) is the number of hours. which expression is equivalent to (p(t))?
2. (300 \cdot 8^t)
3. (300 \cdot 16^t)
4. (300 \cdot 2^t)
5. (300^{2t} \cdot 2^{2t})

1. the growth of a certain organism can be modeled by (c(t) = 10(1.029)^{24t}), where (c(t)) is the total number of cells after (t) hours. which function is approximately equivalent to (c(t))?
2. (c(t) = 240(.083)^{24t})
3. (c(t) = 10(.083)^t)
4. (c(t) = 10(1.986)^t)
5. (c(t) = 240(1.986)^{\frac{t}{24}})

Explanation:

🆕 New Concept Discovered: Laws of Exponents
Rewriting exponential expressions using base and power rules.

Step 1: Solve Question 1

We need to find which function is equivalent to \( f(n) = 6(16)^n \).

Let's rewrite the base \( 16 \) as a power of another number:

$$ 16 = 2^4 $$

Substitute this back into the function:

$$ f(n) = 6(2^4)^n $$

Using the power of a power rule \( (a^m)^n = a^{m \cdot n} \):

$$ f(n) = 6(2)^{4n} $$

This matches option 2: \( h(n) = 6(2)^{4n} \).

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Step 2: Solve Question 2

We need to rewrite the quadratic function \( f(x) = 3x^2 + 12x + 11 \) in vertex form \( f(x) = a(x-h)^2 + k \) by completing the square.

First, group the \( x \)-terms and factor out the leading coefficient \( 3 \):

$$ f(x) = 3(x^2 + 4x) + 11 $$

To complete the square inside the parentheses, take half of the coefficient of \( x \) (which is \( 4 \)), square it to get \( 4 \), and add and subtract it inside:

$$ f(x) = 3(x^2 + 4x + 4 - 4) + 11 $$

$$ f(x) = 3((x + 2)^2 - 4) + 11 $$

Distribute the \( 3 \):

$$ f(x) = 3(x + 2)^2 - 12 + 11 $$

$$ f(x) = 3(x + 2)^2 - 1 $$

This matches option 3: \( f(x) = 3(x+2)^2 - 1 \).

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Step 3: Solve Question 3

We need to find which function is equivalent to \( V = 15,000(0.81)^t \).

Let's rewrite the base \( 0.81 \) as a power:

$$ 0.81 = (0.9)^2 $$

Substitute this back into the equation:

$$ V = 15,000((0.9)^2)^t $$

Using the power of a power rule:

$$ V = 15,000(0.9)^{2t} $$

This matches option 2: \( V = 15,000(0.9)^{2t} \).

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Step 4: Solve Question 4

We need to find which function is equivalent to \( A(t) = 5000(1.013)^{t+25} \).

Using the product rule of exponents \( a^{m+n} = a^m \cdot a^n \):

$$ (1.013)^{t+25} = (1.013)^t \cdot (1.013)^{25} $$

Substitute this back into the expression:

$$ A(t) = 5000(1.013)^t(1.013)^{25} $$

This matches option 4: \( A(t) = 5000(1.013)^t(1.013)^{25} \).

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Step 5: Solve Question 5

We need to find which expression is equivalent to \( P(t) = 300 \cdot 2^{4t} \).

Using the power of a power rule in reverse, \( a^{m \cdot n} = (a^m)^n \):

$$ 2^{4t} = (2^4)^t $$

Evaluate \( 2^4 \):

$$ 2^4 = 16 $$

So:

$$ P(t) = 300 \cdot 16^t $$

This matches option 2: \( 300 \cdot 16^t \).

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Step 6: Solve Question 6

We need to find which function is approximately equivalent to \( C(t) = 10(1.029)^{24t} \).

Using the power of a power rule:

$$ C(t) = 10((1.029)^{24})^t $$

Evaluate \( (1.029)^{24} \) using a calculator:

$$ (1.029)^{24} \approx 1.986 $$

Substitute this back into the expression:

$$ C(t) \approx 10(1.986)^t $$

This matches option 3: \( C(t) = 10(1.986)^t \).

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Answer:

• Question 1: Option 2) \( h(n) = 6(2)^{4n} \)
• Question 2: Option 3) \( f(x) = 3(x+2)^2 - 1 \)
• Question 3: Option 2) \( V = 15,000(0.9)^{2t} \)
• Question 4: Option 4) \( A(t) = 5000(1.013)^t(1.013)^{25} \)
• Question 5: Option 2) \( 300 \cdot 16^t \)
• Question 6: Option 3) \( C(t) = 10(1.986)^t \)