Math / Numbers & Operations

QuestionSolve. $243^{\frac{1}{5}}=?$ \begin{tabular}{l|c|c} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \\ & 0 & \end{tabular}

Studdy Solution

STEP 1

What is this asking? This is asking us to find the fifth root of $243$ , which means finding a number that, when multiplied by itself five times, equals $243$ . Watch out! Don't confuse this with raising $243$ to the power of $5$ .
We're looking for a much smaller number!

STEP 2

1. Prime Factorization
2. Simplify the Expression

STEP 3

Let's break down $243$ into its prime factors.
This means finding the prime numbers that multiply together to give us $243$ .
We can start by dividing $243$ by the smallest prime number, which is $2$ .
Since $243$ is odd, it's not divisible by $2$ .
So, let's try the next smallest prime number, $3$ . $243$ divided by $3$ is $81$ , so we can write $243$ as $3 \cdot 81$ .

STEP 4

Now, we need to factor $81$ .
We know that $81$ is $9$ times $9$ , so we have $243 = 3 \cdot 9 \cdot 9$ .
Since $9$ is $3$ times $3$ , we can write $243$ as $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$ , or $3^5$ .
Awesome!

STEP 5

Now, we can rewrite our original expression using the prime factorization of $243$ .
We have $243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}}$ .

STEP 6

Remember the power of a power rule: $(a^m)^n = a^{m \cdot n}$ .
Let's apply this rule to our expression.
We have $(3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}}$ .

STEP 7

Multiplying the exponents, we get $5 \cdot \frac{1}{5}$ .
This is the same as $\frac{5}{1} \cdot \frac{1}{5} = \frac{5}{5}$ .
Since any number divided by itself is $1$ (except for zero, of course!), we get $3^1$ .

STEP 8

And any number raised to the power of $1$ is just itself!
So, $3^1 = 3$ .
Fantastic!

STEP 9

So, $243^{\frac{1}{5}} = 3$ .
We found that the fifth root of $243$ is $3$ !

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QuestionSolve. $243^{\frac{1}{5}}=?$ \begin{tabular}{l|c|c} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \\ & 0 & \end{tabular}

Studdy Solution

STEP 1

What is this asking? This is asking us to find the fifth root of $243$ , which means finding a number that, when multiplied by itself five times, equals $243$ . Watch out! Don't confuse this with raising $243$ to the power of $5$ .
We're looking for a much smaller number!

STEP 2

1. Prime Factorization
2. Simplify the Expression

STEP 3

STEP 4

Now, we need to factor $81$ .
We know that $81$ is $9$ times $9$ , so we have $243 = 3 \cdot 9 \cdot 9$ .
Since $9$ is $3$ times $3$ , we can write $243$ as $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$ , or $3^5$ .
Awesome!

STEP 5

Now, we can rewrite our original expression using the prime factorization of $243$ .
We have $243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}}$ .

STEP 6

Remember the power of a power rule: $(a^m)^n = a^{m \cdot n}$ .
Let's apply this rule to our expression.
We have $(3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}}$ .

STEP 7

Multiplying the exponents, we get $5 \cdot \frac{1}{5}$ .
This is the same as $\frac{5}{1} \cdot \frac{1}{5} = \frac{5}{5}$ .
Since any number divided by itself is $1$ (except for zero, of course!), we get $3^1$ .

STEP 8

And any number raised to the power of $1$ is just itself!
So, $3^1 = 3$ .
Fantastic!

STEP 9

So, $243^{\frac{1}{5}} = 3$ .
We found that the fifth root of $243$ is $3$ !

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QuestionSolve. 24315=?243^{\frac{1}{5}}=?24351​=? \begin{tabular}{l|c|c} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \\ & 0 & \end{tabular}

Studdy Solution

STEP 1

What is this asking? This is asking us to find the **fifth root** of 243243243, which means finding a number that, when multiplied by itself five times, equals 243243243. Watch out! Don't confuse this with raising 243243243 to the power of 555.We're looking for a much smaller number!

STEP 2

1. Prime Factorization2. Simplify the Expression

STEP 3

STEP 4

Now, we need to factor 818181.We know that 818181 is 999 times 999, so we have 243=3⋅9⋅9243 = 3 \cdot 9 \cdot 9243=3⋅9⋅9.Since 999 is 333 times 333, we can write 243243243 as 3⋅3⋅3⋅3⋅33 \cdot 3 \cdot 3 \cdot 3 \cdot 33⋅3⋅3⋅3⋅3, or 353^535.Awesome!

STEP 5

Now, we can **rewrite** our original expression using the prime factorization of 243243243.We have 24315=(35)15243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}}24351​=(35)51​.

STEP 6

Remember the **power of a power rule**: (am)n=am⋅n(a^m)^n = a^{m \cdot n}(am)n=am⋅n.Let's apply this rule to our expression.We have (35)15=35⋅15(3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}}(35)51​=35⋅51​.

STEP 7

Multiplying the exponents, we get 5⋅155 \cdot \frac{1}{5}5⋅51​.This is the same as 51⋅15=55\frac{5}{1} \cdot \frac{1}{5} = \frac{5}{5}15​⋅51​=55​.Since any number divided by itself is 111 (except for zero, of course!), we get 313^131.

STEP 8

And any number raised to the power of 111 is just itself!So, 31=33^1 = 331=3.Fantastic!

STEP 9

So, 24315=3243^{\frac{1}{5}} = 324351​=3.We found that the fifth root of 243243243 is 333!

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QuestionSolve. $243^{\frac{1}{5}}=?$ \begin{tabular}{l|c|c} 7 & 8 & 9 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \\ & 0 & \end{tabular}

What is this asking? This is asking us to find the fifth root of $243$ , which means finding a number that, when multiplied by itself five times, equals $243$ . Watch out! Don't confuse this with raising $243$ to the power of $5$ .
We're looking for a much smaller number!

1. Prime Factorization
2. Simplify the Expression

Now, we need to factor $81$ .
We know that $81$ is $9$ times $9$ , so we have $243 = 3 \cdot 9 \cdot 9$ .
Since $9$ is $3$ times $3$ , we can write $243$ as $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$ , or $3^5$ .
Awesome!

Now, we can rewrite our original expression using the prime factorization of $243$ .
We have $243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}}$ .

Remember the power of a power rule: $(a^m)^n = a^{m \cdot n}$ .
Let's apply this rule to our expression.
We have $(3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}}$ .

Multiplying the exponents, we get $5 \cdot \frac{1}{5}$ .
This is the same as $\frac{5}{1} \cdot \frac{1}{5} = \frac{5}{5}$ .
Since any number divided by itself is $1$ (except for zero, of course!), we get $3^1$ .

And any number raised to the power of $1$ is just itself!
So, $3^1 = 3$ .
Fantastic!

So, $243^{\frac{1}{5}} = 3$ .
We found that the fifth root of $243$ is $3$ !