Answer :
To find the missing number in the given exponential expression \(7^? \div 7^5 = 7^{12}\), let's follow these steps:
1. Understand the properties of exponents: When you divide two expressions with the same base, you subtract the exponent of the divisor from the exponent of the dividend. Mathematically, this property is expressed as:
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
2. Apply this property to the given problem: In this problem, we are working with the base \(7\). The given expression is:
[tex]\[ \frac{7^?}{7^5} = 7^{12} \][/tex]
3. Set up the equation by using the property: According to the property, the division of \(7^?\) by \(7^5\) can be rewritten with a single exponent:
[tex]\[ 7^{?-5} = 7^{12} \][/tex]
4. Equate the exponents: Because the bases are the same (\(7\)), we can equate the exponents from both sides of the equation:
[tex]\[ ? - 5 = 12 \][/tex]
5. Solve for the missing exponent: To find the value of \(?\), solve the equation:
[tex]\[ ? - 5 = 12 \][/tex]
Add \(5\) to both sides of the equation:
[tex]\[ ? = 12 + 5 \][/tex]
[tex]\[ ? = 17 \][/tex]
Therefore, the missing number in the expression [tex]\(7^? \div 7^5 = 7^{12}\)[/tex] is [tex]\(17\)[/tex].
1. Understand the properties of exponents: When you divide two expressions with the same base, you subtract the exponent of the divisor from the exponent of the dividend. Mathematically, this property is expressed as:
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
2. Apply this property to the given problem: In this problem, we are working with the base \(7\). The given expression is:
[tex]\[ \frac{7^?}{7^5} = 7^{12} \][/tex]
3. Set up the equation by using the property: According to the property, the division of \(7^?\) by \(7^5\) can be rewritten with a single exponent:
[tex]\[ 7^{?-5} = 7^{12} \][/tex]
4. Equate the exponents: Because the bases are the same (\(7\)), we can equate the exponents from both sides of the equation:
[tex]\[ ? - 5 = 12 \][/tex]
5. Solve for the missing exponent: To find the value of \(?\), solve the equation:
[tex]\[ ? - 5 = 12 \][/tex]
Add \(5\) to both sides of the equation:
[tex]\[ ? = 12 + 5 \][/tex]
[tex]\[ ? = 17 \][/tex]
Therefore, the missing number in the expression [tex]\(7^? \div 7^5 = 7^{12}\)[/tex] is [tex]\(17\)[/tex].
