Math / Algebra

QuestionWork out the value of $w$ in this equality: $9^{3} \times 9^{2} \times 9^{w}=9^{42}$

Studdy Solution

STEP 1

1. The equation involves exponential expressions with the same base.
2. We can use properties of exponents to simplify and solve for $w$ .

STEP 2

1. Simplify the left side of the equation using properties of exponents.
2. Equate the exponents on both sides of the equation.
3. Solve for $w$ .

STEP 3

Use the property of exponents that states $a^m \times a^n = a^{m+n}$ to simplify the left side of the equation:
$9^3 \times 9^2 \times 9^w = 9^{3+2+w}$
This simplifies to:
$9^{3+2+w} = 9^{42}$

STEP 4

Since the bases are the same, set the exponents equal to each other:
$3 + 2 + w = 42$

STEP 5

Solve for $w$ by combining like terms and isolating $w$ :
$5 + w = 42$
Subtract 5 from both sides:
$w = 42 - 5$ $w = 37$
The value of $w$ is:
$\boxed{37}$

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QuestionWork out the value of $w$ in this equality: $9^{3} \times 9^{2} \times 9^{w}=9^{42}$

Studdy Solution

STEP 1

1. The equation involves exponential expressions with the same base.
2. We can use properties of exponents to simplify and solve for $w$ .

STEP 2

1. Simplify the left side of the equation using properties of exponents.
2. Equate the exponents on both sides of the equation.
3. Solve for $w$ .

STEP 3

Use the property of exponents that states $a^m \times a^n = a^{m+n}$ to simplify the left side of the equation:
$9^3 \times 9^2 \times 9^w = 9^{3+2+w}$
This simplifies to:
$9^{3+2+w} = 9^{42}$

STEP 4

Since the bases are the same, set the exponents equal to each other:
$3 + 2 + w = 42$

STEP 5

Solve for $w$ by combining like terms and isolating $w$ :
$5 + w = 42$
Subtract 5 from both sides:
$w = 42 - 5$ $w = 37$
The value of $w$ is:
$\boxed{37}$

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QuestionWork out the value of www in this equality: 93×92×9w=9429^{3} \times 9^{2} \times 9^{w}=9^{42}93×92×9w=942

Studdy Solution

STEP 1

1. The equation involves exponential expressions with the same base.2. We can use properties of exponents to simplify and solve for w w w.

STEP 2

1. Simplify the left side of the equation using properties of exponents.2. Equate the exponents on both sides of the equation.3. Solve for w w w.

STEP 3

Use the property of exponents that states am×an=am+n a^m \times a^n = a^{m+n} am×an=am+n to simplify the left side of the equation:93×92×9w=93+2+w 9^3 \times 9^2 \times 9^w = 9^{3+2+w} 93×92×9w=93+2+wThis simplifies to:93+2+w=942 9^{3+2+w} = 9^{42} 93+2+w=942

STEP 4

Since the bases are the same, set the exponents equal to each other:3+2+w=42 3 + 2 + w = 42 3+2+w=42

STEP 5

Solve for w w w by combining like terms and isolating w w w:5+w=42 5 + w = 42 5+w=42Subtract 5 from both sides:w=42−5 w = 42 - 5 w=42−5 w=37 w = 37 w=37 The value of w w w is:37 \boxed{37} 37​

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QuestionWork out the value of $w$ in this equality: $9^{3} \times 9^{2} \times 9^{w}=9^{42}$

1. The equation involves exponential expressions with the same base.
2. We can use properties of exponents to simplify and solve for $w$ .

1. Simplify the left side of the equation using properties of exponents.
2. Equate the exponents on both sides of the equation.
3. Solve for $w$ .

Use the property of exponents that states $a^m \times a^n = a^{m+n}$ to simplify the left side of the equation:
$9^3 \times 9^2 \times 9^w = 9^{3+2+w}$
This simplifies to:
$9^{3+2+w} = 9^{42}$

Since the bases are the same, set the exponents equal to each other:
$3 + 2 + w = 42$

Solve for $w$ by combining like terms and isolating $w$ :
$5 + w = 42$
Subtract 5 from both sides:
$w = 42 - 5$ $w = 37$
The value of $w$ is:
$\boxed{37}$