Express as a power of 3 in 729 and 343.

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Vedantu Content Team · Accepted Answer

Hint: Here, we need to express 729 and 343 as a power of 3. We will write the given numbers as a product of their factors. Then, we will simplify the product using rules of exponents, and express the numbers as a power of 3.Formula Used: We will use the rule of exponent: If two or more numbers with same base and different exponents are multiplied, the product can be written as $${a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}$$.Complete step-by-step answer:We will first find the factor of 729.Let us use the divisibility tests of 2, 3, 5, etc. to find the factors of 729.First, we will check the divisibility by 2.We know that a number is divisible by 2 if it is an even number.This means that any number that has one of the digits 2, 4, 6, 8, or 0 in the unit’s place, is divisible by 2.We can observe that the number 729 has 9 at the unit’s place. Therefore, the number 729 is not divisible by 2.Next, we will check the divisibility by 3.A number is divisible by 3 only if the sum of its digits is divisible by 3.We will add the digits of the number 729.Thus, we get$$7 + 2 + 9 = 18$$Since the number 18 is divisible by 3, the number 729 is divisible by 3.Dividing 729 by 3, we get$$\dfrac{{729}}{3} = 243$$Next, we will add the digits of the number 243.Thus, we get$$2 + 4 + 3 = 9$$Since the number 9 is divisible by 3, the number 243 is divisible by 3.Dividing 243 by 3, we get$$\dfrac{{243}}{3} = 81$$Next, we will add the digits of the number 81.Thus, we get$$8 + 1 = 9$$Since the number 9 is divisible by 3, the number 81 is divisible by 3.Dividing 81 by 3, we get$$\dfrac{{81}}{3} = 27$$We know that 27 is the product of 3 and 9.Therefore, we can rewrite the number 729 as$$729 = 3 \times 3 \times 3 \times 3 \times 9$$Multiplying the pairs of 3 with each other, we get$$\begin{array}{l} \Rightarrow 729 = 9 \times 9 \times 9\\ \Rightarrow 729 = {9^1} \times {9^1} \times {9^1}\end{array}$$By using rule of exponent, $${a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}$$, we can rewrite 729 as$$\begin{array}{l} \Rightarrow 729 = {9^{1 + 1 + 1}}\\ \Rightarrow 729 = {9^3}\end{array}$$$$\therefore $$ We have expressed 729 as 9 raised to the power 3.Now, let us use the divisibility tests of 2, 3, 5, etc. to find the factors of 343.First, we will check the divisibility by 2.We know that a number is divisible by 2 if it is an even number.This means that any number that has one of the digits 2, 4, 6, 8, or 0 in the unit’s place, is divisible by 2.We can observe that the number 343 has 3 at the unit’s place. Therefore, the number 343 is not divisible by 2.Next, we will check the divisibility by 3.A number is said to be divisible by 3 only if the sum of its digits is divisible by 3.We will add the digits of the number 343.Thus, we get$$3 + 4 + 3 = 10$$Since the number 10 is not divisible by 3, the number 343 is not divisible by 3.Now, we will check the divisibility by 5.We know that a number that has one of the digits 0 or 5 in the unit’s place, is divisible by 5.We can observe that the number 343 has 3 at the unit’s place. Therefore, the number 343 is not divisible by 5.Now, we will check the divisibility by 7.If double the digit at unit’s place is subtracted from the remaining number, and the result is divisible by 7, then the number is divisible by 7.The digit at the unit's place in 343 is 3. The remaining number is 34. The double of the digit at the unit's place is $$3 \times 2 = 6$$.Subtracting 6 from 34, we get$$34 - 6 = 28$$Since the number 28 is divisible by 7, the number 343 is divisible by 7.Dividing 343 by 7, we get$$\dfrac{{343}}{7} = 49$$We know that 49 is the product of 7 and 7.Therefore, we can rewrite the number 343 as$$\begin{array}{l}343 = 7 \times 7 \times 7\\ \Rightarrow 343 = {7^1} \times {7^1} \times {7^1}\end{array}$$Using the rules of exponents, we get$$\begin{array}{l} \Rightarrow 343 = {7^{1 + 1 + 1}}\\ \Rightarrow 343 = {7^3}\end{array}$$$$\therefore $$ We have expressed 343 as 7 raised to the power 3.Note:  We expressed 729 as 9 raised to the power 3, and 343 as 7 raised to the power 3. Here, 729 is called the cube of 9. Similarly, 343 is the cube of 7. When a number is raised to power 3, the resulting product is its cube, whereas when a number is raised to power 2 then the resulting power is called its square.