Solving Exponential and Radical Equations

The problem asks us to find the value of 'm' in the equation \( \mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}} \). To solve this, we need to simplify the right-hand side (RHS) of the equation by converting the radicals (roots) into exponents. Remember the rules for exponents and radicals: \( \sqrt[n]{\mathbf{a}} = \mathbf{a}^{1/\mathbf{n}} \) and \( \mathbf{a}^{\mathbf{p}} \cdot \mathbf{a}^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}+\mathbf{q}} \) and \( (\mathbf{a}^{\mathbf{p}})^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}\mathbf{q}} \).

Step-by-Step Simplification of the Right-Hand Side

Let's simplify the expression inside the fourteenth root step by step, working from the innermost radical outwards.

1. Innermost term: We have \( \sqrt{\mathbf{x}} \). Using the rule \( \sqrt[n]{\mathbf{a}} = \mathbf{a}^{1/\mathbf{n}} \) with \( \mathbf{n}=2 \) (square root), this becomes \( \mathbf{x}^{1/2} \).
2. Next term: The expression becomes \( \mathbf{x} \sqrt{\mathbf{x}} \), which is \( \mathbf{x} \cdot \mathbf{x}^{1/2} \). Using the rule \( \mathbf{a}^{\mathbf{p}} \cdot \mathbf{a}^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}+\mathbf{q}} \), this is \( \mathbf{x}^{1 + 1/2} = \mathbf{x}^{3/2} \).
3. Next radical: We have \( \sqrt{\mathbf{x} \sqrt{\mathbf{x}}} \), which is \( \sqrt{\mathbf{x}^{3/2}} \). Using the rule \( \sqrt[n]{\mathbf{a}} = \mathbf{a}^{1/\mathbf{n}} \) with \( \mathbf{n}=2 \), this is \( (\mathbf{x}^{3/2})^{1/2} \). Using the rule \( (\mathbf{a}^{\mathbf{p}})^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}\mathbf{q}} \), this simplifies to \( \mathbf{x}^{(3/2) \times (1/2)} = \mathbf{x}^{3/4} \).
4. Next term: The expression inside the fourteenth root is \( \mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}} \), which is \( \mathbf{x} \cdot \mathbf{x}^{3/4} \). Using the rule \( \mathbf{a}^{\mathbf{p}} \cdot \mathbf{a}^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}+\mathbf{q}} \), this is \( \mathbf{x}^{1 + 3/4} = \mathbf{x}^{7/4} \).
5. Outermost radical: We have \( \sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}} \), which is \( \sqrt[14]{\mathbf{x}^{7/4}} \). Using the rule \( \sqrt[n]{\mathbf{a}} = \mathbf{a}^{1/\mathbf{n}} \) with \( \mathbf{n}=14 \), this is \( (\mathbf{x}^{7/4})^{1/14} \). Using the rule \( (\mathbf{a}^{\mathbf{p}})^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}\mathbf{q}} \), this simplifies to \( \mathbf{x}^{(7/4) \times (1/14)} \).
6. Final exponent calculation: \( (7/4) \times (1/14) = 7 / (4 \times 14) = 7 / 56 \). We can simplify the fraction \( 7/56 \) by dividing the numerator and denominator by 7: \( 7 \div 7 = 1 \) and \( 56 \div 7 = 8 \). So, the exponent is \( 1/8 \).

Thus, the right-hand side of the equation simplifies to \( \mathbf{x}^{1/8} \).

Equating the Exponents

The original equation is \( \mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}} \). We have simplified the RHS to \( \mathbf{x}^{1/8} \). So the equation becomes:

\( \mathbf{x}^{\mathbf{m}} = \mathbf{x}^{1/8} \)

If the bases are the same (in this case, the base is \( \mathbf{x} \)), and the two expressions are equal, then the exponents must be equal.

Therefore, \( \mathbf{m} = 1/8 \).

Conclusion

By simplifying the expression involving radicals and exponents on the right-hand side of the equation \( \mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}} \), we found that it is equal to \( \mathbf{x}^{1/8} \). By comparing the exponents on both sides, we determined the value of m.

Revision Table: Exponent and Radical Rules

Review these fundamental rules used in solving the problem:

• Product of powers with same base: \( \mathbf{a}^{\mathbf{p}} \cdot \mathbf{a}^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}+\mathbf{q}} \)
• Power of a power: \( (\mathbf{a}^{\mathbf{p}})^{\mathbf{q}} = \mathbf{a}^{\mathbf{p}\mathbf{q}} \)
• nth root as exponent: \( \sqrt[n]{\mathbf{a}} = \mathbf{a}^{1/\mathbf{n}} \)
• Square root as exponent: \( \sqrt{\mathbf{a}} = \mathbf{a}^{1/2} \)

Additional Information: Solving Exponential Equations

An exponential equation is an equation where the variable appears in the exponent. A common strategy to solve simple exponential equations is to make the bases on both sides of the equation the same. Once the bases are the same, you can equate the exponents.

For example, if you have \( \mathbf{a}^{\mathbf{f}(\mathbf{x})} = \mathbf{a}^{\mathbf{g}(\mathbf{x})} \), where \( \mathbf{a} \) is a positive number other than 1, then you can conclude that \( \mathbf{f}(\mathbf{x}) = \mathbf{g}(\mathbf{x}) \). In our problem, we had \( \mathbf{x}^{\mathbf{m}} = \mathbf{x}^{1/8} \), where the base is \( \mathbf{x} \) and the exponents are \( \mathbf{m} \) and \( 1/8 \). Assuming \( \mathbf{x} > 0 \) and \( \mathbf{x} \neq 1 \), we could equate the exponents \( \mathbf{m} = 1/8 \).