how can we rewrite and/ or simplify expressions in exponential form?

To solve this problem, the two consecutive odd integers are defined as x and x+2. A mathematical equation is created according to the problem conditions and solved. The consecutive odd integers are found to be 81 and 83.

Let's define the two consecutive odd integers as x (the smaller) and x+2 (the larger). Given the condition from the question that the sum of the smaller integer and three times the larger integer is 330, we can form the equation x + 3*(x+2) = 330.

We'll simplify that equation, which gives us: x + 3x + 6 = 330, and combining like terms, we get 4x + 6 = 330. We can solve for x by subtracting 6 from both sides and then dividing both sides by 4, which results in x = 81. Given that x represents the smaller odd integer, the larger integer would be x + 2 or 83.

So, the two consecutive odd integers fulfilling the conditions given are 81 and 83.

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The two consecutive odd integers such that the sum of the smaller and 3 times the larger is 330 are 81 and 83.

Denote the smaller odd integer as x and the larger consecutive odd integer as x + 2.

According to the problem, the sum of the smaller integer and 3 times the larger integer is 330. So, we have the equation:

x + 3(x + 2) = 330

Now, solve for x :

1. Distribute the 3 on the left side:

x + 3x + 6 = 330

2. Combine like terms:

4x + 6 = 330

3. Subtract 6 from both sides to isolate terms with x:

4x = 324

4. Divide both sides by 4 to solve for x:

[tex]\[ x = \frac{324}{4} \]\[ x = 81 \][/tex]

So, the smaller odd integer x is 81. Now find the larger consecutive odd integer:

x + 2 = 81 + 2 = 83

Therefore, the two consecutive odd integers are 81 and 83.

To verify:

- The sum of the smaller integer and 3 times the larger integer:

[tex]\[ 81 + 3 \cdot 83 = 81 + 249 = 330 \][/tex]

The verification confirms that the solution is correct.

So, the two consecutive odd integers such that the sum of the smaller and 3 times the larger is 330 are 81 and 83.

The function is increasing in:

                  x< -1

and it is decreasing in:

            x > -1

and nowhere constant.

After accounting for the $5 processing fee, you can purchase approximately 5 concert tickets with $100, as each ticket cost $18.

To determine how many concert tickets you can buy for $100, we need to account for both the fixed processing fee and the variable cost per ticket. The processing fee is a one-time payment, and the cost per ticket is multiplied by the number of tickets you want to purchase. We can set up the following equation where T represents the number of tickets:

$5 + $18T = $100

First, we subtract the processing fee from the total amount of money available:

$100 - $5 = $95

Now we divide the remaining amount by the cost per ticket:

$95 / $18 = 5.28

Since you cannot buy a fraction of a ticket, you would be able to purchase 5 tickets with $100, with a remaining balance of $5 (since 5 tickets cost $90, plus the $5 processing fee).

Answer:

The greatest number of identical packages Shaquira can make is 21.

Explanation:

To find the greatest number of identical packages Shaquira can make, we need to determine the greatest common divisor (GCD) of the number of chocolate chip cookies and the number of sugar cookies. The GCD represents the largest number of identical packages that can be made.

Given:

- Chocolate chip cookies: 86

- Sugar cookies: 42

We can find the GCD using the Euclidean algorithm or by listing the factors of each number and identifying the largest common factor.

By listing the factors:

- Factors of 86: 1, 2, 43, 86

- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The common factors are 1 and 2. The greatest common factor (GCD) is 2.

So, Shaquira can make the greatest number of identical packages by dividing both the number of chocolate chip cookies and the number of sugar cookies by the GCD:

[tex]\( \frac{86}{2} = 43 \)[/tex] packages from chocolate chip cookies

[tex]\( \frac{42}{2} = 21 \)[/tex] packages from sugar cookies

In the number 123,456,789, the digit with the greatest value is 1.

The explanation is as given:

Each digit contributes value based on its position in the number. The leftmost digit holds the highest place value.

In this case, the digit 1 is in the hundreds of millions place.

1 (in the hundreds of millions place) = 1 x 100,000,000 = 100,000,000

Although the digit 9 appears at the end, it's in the ones place, which has the lowest weight (9 x 1 = 9).

Therefore, the digit with the greatest value is 1, and its value is 100,000,000.

Answer:

With 16.32 square yards, James gets 12 patches.

With 32.64 square yards, he gets 6 patches.

Step-by-step explanation:

James has 195.84 square yards of land he can use to grow vegetables.

He wants to divide this into patches of 16.32 square yards.

So, he will get [tex]\frac{195.84}{16.32}=12[/tex] equal patches.

If he wants to create garden patches with an area of 32.64 square yards, then he gets [tex]\frac{195.84}{32.64}=6[/tex] patches.

Answer:

[tex]2.25\times 10^1[/tex] times greater is the weight of the elephant than the weight of the lion.

Step-by-step explanation:

Given : A zookeeper weighed an African elephant to be [tex]9\times 10^3[/tex] pounds and an African lion to be [tex]4\times 10^2[/tex] pounds.  

To find : How many times greater is the weight of the elephant than the weight of the lion?

Solution :  

A zookeeper weighed an African elephant to be [tex]9\times 10^3[/tex] pounds.

A zookeeper weighed an African lion to be [tex]4\times 10^2[/tex] pounds

Now, Divide the weighed an African elephant with weighed an African lion to determine times greater is the weight of the elephant than the weight of the lion,

[tex]n=\frac{9\times 10^3}{4\times 10^2}[/tex]

[tex]n=2.25\times 10^1[/tex]

Therefore, Option A is correct.

[tex]2.25\times 10^1[/tex] times greater is the weight of the elephant than the weight of the lion.

Let us say that:

s = speed of Donna

s + 2 = speed of April

Then we are also given that:

t + (25/60) = time of Donna

t = time of April

The distance taken by the two should be equal. We know that distance is the product of speed and time, so:

s * (t + 25/60) = (s + 2) t

We also know that:

(s + 2) * t = 5

s = 5/t – 2

So that:

(5/t – 2) * (t + 25/60) = 5

5 + 125/60t – 2t – 50/60 = 5

25/12t – 2t – 5/6 = 0

Multiply everything by t:

- 2t^2 – 5/6 t + 25/12 = 0

t^2 + 5/12 t = 25/24

Completing the square:

(t + 5/24)^2 = 25/24 + (5/24)^2

t + 5/24 = ± 1.04

t = -1.25 h, 0.83 h

Since time cannot be negative, therefore t = 0.83 h

So,

(s + 2) * t = 5

s + 2 = 5/0.83 = 6 miles per hour

s = 4 miles per hour

Answers:

speed of Donna = 4 miles / hr

speed of April = 6 miles / hr

80% as a fraction in reduced form is 4/5.

To express 80% as a fraction in reduced form, you need to convert the percentage to a fraction. The term reduced form means simplifying the fraction to its lowest terms.

Here's a step-by-step explanation:

Step 1: Understand what a percentage represents

A percentage is a way to express a fraction or ratio out of 100. So, when you see 80%, it means 80 out of 100.

Step 2: Write the percentage as a fraction

To convert a percentage to a fraction, you can write it as a fraction with the percentage as the numerator and 100 as the denominator. Therefore, 80% can be written as 80/100.

Step 3: Simplify the fraction

To reduce the fraction to its lowest terms, you need to find the greatest common divisor (GCD) between the numerator and denominator. In this case, both 80 and 100 can be divided by 20. Dividing both the numerator and denominator by 20, we get:

80/100 = (80 ÷ 20) / (100 ÷ 20) = 4/5.

Step 4: Interpret the reduced fraction

The fraction 4/5 is the reduced form of 80%. It means that 80% is equivalent to four-fifths (4 parts out of 5).

Therefore, 80% as a fraction in reduced form is 4/5.

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5x - good ones

4x - bad ones

[tex] 4x=36\\ x=9\\\\ 5x=5\cdot9=45\\\\ 45+36=81 [/tex]

81

The change in the width of the beachfront from low to high tide is 102 feet, and the distance from the ocean to the homes at high tide is 248 feet.

a. To find the change in width from low to high tide, we multiply the rate of change (-17 feet/hour) by the time it takes for the ocean to move from low to high tide (6 hours).

So, -17 feet/hour * 6 hours = -102 feet. Therefore, the width of the beachfront decreases by 102 feet from low to high tide.

b. At high tide, the distance from the ocean to the homes is the sum of the original distance (350 feet) and the change in width during the tide change (-102 feet), resulting in 350 feet + (-102 feet) = 248 feet.